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Jerry L. Jensen, Patrick W. M. Corbett, Patrick W. M. Corbett - Statistics for Petroleum Engineers and Geoscientists-Prentice Hall (1997)

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Statistics for
Petroleum Engineers
and eoscientists
Prentice Hall Petroleum Engineering Series
Larry W. Lake, Editor
Economides, Hill, and Economides, Petroleum Production Systems
Jensen, Lake, Corbett, and Goggin, Statistics for Petroleum
Engineers and Geoscientists
Lake, Enhanced Oil Recovery
Laudon, Principles of Petroleum Development Geology
Raghavan, Well Test Analysis
Rogers, Coalbed Methane
Other Titles of Interest
Carcoana, Applied Enhanced Oil Recovery
Craft, Hawkins and Terry, Applied Petroleum Reservoir Engineering
Schecter, Oil Well Stimulation
Whittaker, Mud Logging Handbook
Library of Congress Cataloging-in-Publication Data
Statistics for petroleum engineers and geoscientists I by Jerry L.
Jensen ... [et al.].
p. em.
Includes bibliographical references and index.
ISBN 0-13-131855-1
1. Statistics 2. Geology-Statistical methods. I. Jensen,
Jerry L.
QA276.S788 1997
519.5-dc20
96-33321
CIP
Editorial/production supervision: Eileen Clark
Cover design: Lido Graphics
Manufacturing manager: Alexis Heydt
Acquisitions editor: Bernard Goodwin
Cover director: Jerry Votta
1997 by Prentice Hall PTR
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DEDICATIONS
Dedicated to the glory of God and to my aunt and late
uncle, Joyce and Jack Luck .. .J.L.J
In memory of my mother, Ina B. Lake, 1923-1994 ...
L.W.L.
To my parents for my start in life ... P.W.M.C
To Holly, Amanda, and Janet-- My support and
inspiration ... D.J.G.
CONTENTS
v
Dedications
1
Figures
xiii
Preface
xix
Acknowledgments
xxi
Introduction
1-1
1-2
1-3
1-4
1-5
1-6
2
2
7
8
12
17
18
Basic Concepts
2-1
2-2
2-3
2-4
2-5
2-6
2-7
3
The Role of Statistics In Engineering And
Geology
The Measurements
The Medium
The Physics Of Flow
Estimation And Uncertainty
Summary
1
Combinations Of Events
Probability
Probability Laws
Conditional Probabilities
Independence
Bayes' Theorem
Summary
19
21
23
25
27
29
32
36
Univariate Distributions
3-1
The Cumulative Distribution Function
37
41
vii
Contents
viii
3-2
3-3
3-4
3-5
4
Noncentered Moments
Centered Moments
Combinations And Other Moments
Common PDF's
Truncated Data Sets
Partitioning Data Sets
Closing Comment
Estimators
5-1
5-2
5-3
5-4
5-5
5-6
5-7
5-8
5-9
5-10
5-11
5-12
5-13
5-14
5-15
5-16
5-17
5-18
5-19
6
49
61
61
64
Features Of Single- Variable Distributions 67
4-1
4-2
4-3
4-4
4-5
4-6
4-7
5
Order Statistics
Functions Of Random Variables
The Monte Carlo Method
Sampling And Statistics
67
71
75
76
93
98
105
And Their Assessment
The Elements Of Estimation
Desirable Properties Of Estimators
Estimator Bias
Estimator Efficiency
Estimator Robustness
Assessing The Sensitivities Of Estimators
To Erroneous Data
Sensitivity Coefficients
Confidence Intervals
Computational Methods To Obtain
Confidence Intervals
Properties Of The Sample Mean
Properties Of The Sample Variance
Confidence Intervals For The Sample Mean
Properties Of The Sample Median
Properties Of The Interquartile Range
Estimator For Nonarithmetic Means
Flow Associations For Mean Values
Physical Interpretability In Estimation
Weighted Estimators
Summary Remarks
107
108
109
109
111
115
116
118
122
124
128
129
130
133
134
134
135
138
139
141
Measures Of Heterogeneity
6-1
6-2
Definition Of Heterogeneity
Static Measures
143°
144
144
ix
Statistics for Petroleum Engineers and Geoscientists
6-3
6-4
7
8-1
8-3
8-4
8-5
9-6
9-7
9-8
9-9
9-10
9-11
9-12
9-13
10
170
181
188
Correlation
Joint Distributions
The Covariance Of X And Y
The Joint Normal Distribution
The Reduced Major Axis (Rma) And
Regression Lines
Summary Remarks
Bivariate Analysis:
9-1
9-2
9-3
9-4
9-5
167
Parametric Hypothesis Tests
Nonparametric Tests
Summary Remarks
Bivariate Analysis:
8-2
9
. 162
165
Hypothesis Tests
7-1
7-2
7-3
8
Dynamic Measures
Summary Remarks
189
189
192
195
199
203
Linear Regression
The Elements Of Regression
The Linear Model
The Least-Squares Method
Properties Of Slope And Intercept Estimates
Separately Testing The Precision Of The
Slope And Intercept
Jointly Testing The Precision Of The Slope
And Intercept
Confidence Intervals For The Mean
Response Of The Linear Model And New
Observations
Residuals Analysis
The Coefficient Of Determination
Significance Tests Using The Analysis Of
Variance
The No-Intercept Regression Model
Other Considerations
Summary Remarks
205
206
207
209
213
216
220
223
224
226
228
230
231
233
Bivariate Analysis: Further Linear
Regression
235
Contents
X
10-1
10-2
10-3
10-4
10-5
10-6
10-7
10-8
11
Alternatives To The Least-Squares Lines
Variable Transfonnations
Weighted Least Squares
Reducing The Number Of Variables
The X-A:xis Intercept And Error Bounds
Improving Correlations For Better Prediction
Multivariate Regression
Concluding Comments
Data Types
Nominal Variables
Measures Of Autocorrelation
Estimating Autocorrelation Measures
Semivariograms In Hypothetical Media
Semivariograms In Real Media
Bias And Precision
Models
The Impact Of Autocorrelation Upon The
Information Content Of Data
11-10 Concluding Remarks
265
266
267
273
276
281
285
287
290
292
293
Modeling Geological Media
12-1
12-2
12-3
12-4
12-5
12-6
13
263
Analysis Of Spatial Relationships
11-1
11-2
11-3
11-4
ll-5
11-6
11-7
11-8
11-9
12
235
239
245
246
249
253
255
260
Statistical Point OfView
Kriging
Other Kriging Types
Conditional 'Simulation
Simulated Annealing
Concluding Remarks
294
294
310
312
318
326
The Use Of Statistics In Reservoir
Modeling
13-1
13-2
13-3
13-4
13-5
The Importance Of Geology
Analysis And Inference
Flow Modeling Results
Model Building
Concluding Remarks
327
328
334
342
352
358
Statistics for Petroleum Engineers and Geoscientists
xi
Nomenclature
359
References
367
Index
383
IG
1-1
1-2
1-3
1-4
2-1
2-2
2-3
3-1
3-2
3-3
3-4
3-5
3-6
3-7
3-8
3-9
3-10
3-11
3-12
3-13
3-14
3-15
4-1
4-2
4-3
4-4
ES
The outcome of a flipped coin
Bifurcation diagram for May's equation
Lamination, bed, and bedset scales of sedimentary structure with schematic
permeability variations
Shallow marine stratal elements and measurement scales
Sketch of an experiment, measuring the porosity ( ¢) of a core-plug rock
sample
Hypothetical wireline measurement patterns for three reservoir facies
Probability tree for Example 7
The random variable X assigns a real number to each possible outcome w
An example CDF for a discrete random variable
An example CDF for a continuous random variable
Shale size CDF's based on outcrop data
Empirical CDF's for the data of Example 4a
Example of a PDF for a discrete random variable
Example of a PDF for a continuous random variable
PDF's for the data and log-transformed data of Example 4b
Example histograms for identical data sets with different class sizes
Geological sectiqn and histograms of core porosity and permeability for
the Lower Brent Group
Porosity and permeability histograms for the Etive and Rannoch
formations in the Lower Brent
Transformation from random variable X to Y with change in CDF
Transformation for first random number in Example 7
Schematic procedure for producing STOIIP estimates using Monte Carlo
Reservoir property distributions at sampled ag_d unsampled locations
Probability distribution functions for R and ¢ for I= 2 samples and
Prob(¢ = 1) = 0.5
Probability distribution functions for R and If) for I= 6 samples and
Prob(¢ = 1) = 0.5
The progression of a binomial PDF to a normal distribution for p = 0.9
The continuous normal PDF and CDF
4
6
11
12
20
22
31
38
43
44
46
48
50
51
53
55
57
58
60
61
64
66
78
78.
81
83
xiii
Figures
xiv
4-5
4-6
4-7
4-8
4-9
4-10
4-11
4-12
4-13
4-14
4-15
4-16
4-17
4-18
5-1
5-2
5-3
5-4
5-5
5-6
5-7
5-8
5-9
5-10
5-11
6-1
6-2
6-3.
6-4.
6-5.
Probability coordinates
PDF and CDF for a log-normal distribution
PDF shapes for various p
PDF and CDF for a uniform distribution
PDF and CDF for a triangular distribution
PDF and CDF for an exponential distribution
Schematic plots of truncation (upper) and mixed populations (lower)
Schematic of the effects of truncation on a log-normal CDF
CDF for original oil in place showing fb in percent.
CDF illustrating correction for top truncation
CDF's for nonintersecting parent populations
Experimental CDF for Example 7
CDF's for intersecting populations
"Thumbprint" CDF's for truncated and mixed populations
Estimator bias (b) depends upon the PDF of()
Estimator efficiency is based on how variable the estimates ~are
Uniform PDF with lower limit 0, upper limit 8, and mean 8/2
PDF's of ~1 and ~2 with thex axis scaled in units of (8/3!)112, centered at 8
The true value for 8 is not known. The size of interval where it may be
located depends on the probability assignment
Porosity data probability plot showing approximately normal PDF
The Student's distributio_!!s change shape with the degrees of freedom (df)
Confidence limits for X
Effective permeability for layer-parallel flow
Effective permeability for layer-transverse flow
Porosity measurements from four locations in a reservoir
A
Cv variability(/= 25 samples) for normal and log-normal populations
Variation of Cv with scale and depositional environment
Core-plug and probe-permeameter data in a Rannoch formation interval
Dykstra-Parsons plot
Variability performance of VDP estimator
84
88
89
90
92
93
94
95
96
98
100
103
104
105
110
111
112
123
123
127
132
132
136
137
139
146
147
153
154
156
A
6-6.
6-7
6-8
6-9
6-10
6-11
7-1
Probability plot for VDP in Example 3
Lorenz plot for heterogeneity
Bias performance of the Lorenz coefficient estimator
Variability of Lorenz coefficient estimates
Koval factor and Dykstra-Parsons coefficient for a uniformly layered
medium
Relationship between Koval factor and Gelhar-Axness coefficient
Two arithmetic averages and their 95% confidence intervals (twice their
standard errors)
157
159
161
161
164
164
168
Statistics for Petroleum Engineers and Geoscientists
7-2
7-3
7-4
7-5
7-6
8-1
8-2
8-3
8-4
8-5
8-6
8-7
8-8
8-9
9-1
9-2
9-3
9-4
9-5
9-6
9-7
9-8
9-9
9-10
10-1
10-2
XV
Two samples with differing average permeabilities but similar flow
properties
169
Relationships of averages and their 95% confidence limits
173
Confidence intervals for two Dykstra-Parsons coefficient estimates
178
Comparison of two CDF's in the Kolmogorov-Smirnov test
185
Empirical CDF's for Example 5
187
A mixed joint PDF for X discrete and Y continuous, with their bimodal
marginal PDF's
190
A plane with constantf(x, y) represents (X, Y) pairs with a specific
probability of occurrence
191
Standardized variables x* and y* shift the original X-Y relationship (left)
to the origin and produce a change of scales
194
Bivariate data sets for Example 1
195
A joint normal PDF with p =0.7
196
Two data-set scatter plots with normal probability ellipses
197
JND ellipses of X and Y and their RMA and regression lines
200
Different regression lines can be determined from a data set, depending
upon which distances are minimized
201
Porosity (X)-permeability (Y) scatter plot and lines for Example 3
202
Many data in an X-Y scatter plot (left) provide directly for an X-Y
relationship
207
The least-squares procedure determines a line that minimizes the sum of all
the squared differences, (Yi- Yi)2, between the data points and the line
209
The least-squares procedure finds the minimum value of S.
210
Wireline and core-plug porosities with regression line
217
Diagnostic probability (top left) and residuals plots (top right and bottom)
suggest the errors are normally distributed, show no systematic variation
over the range of X, and are uncorrelated
219
Confidence regions for f3o and [3 1, as reflected by the data, for independent
assessments (left) and joint assessment (right) are shown by the shaded
regions. (C. I. = confidence interval.)
221
The rectangular confidence region obtained from independent assessments
of f3o and /3 1 and the joint, elliptical confidence region of o and [3 1
222
While a linear model (left) appears suitable, a residual plot suggests
otherwise
225
Two data sets with near-zero regression line slopes but different
explanatory powers of X
226
Rannach fonnation porosity-permeability data and regression lines
including (left) and excluding (right) the carbonate concretions
228
Regression line and data (left) and layered formation (right)
238
Regression residuals showing an inappropriate noise model
240
"
/\
A "
xvi
10-3
10-4
10-5
10-6
10-7
10-8
10-9
10-10
11-1
11-2
11-3
11-4
11-5
11-6
ll-7A
ll-7B
ll-8
11-9
11-10
11-11
11-12
11-13
11-14
11-15
12-1
12-2
12-3
12-4
12-5
Figures
The regression-line procedure gives equal weight to deviations of 9,000 and
900
Data set for Example 2 with log- and untransformed plots
Scatter plots and regression lines for Y and log(Y)
Histograms for the data in Fig. 10-5
Histogram, scatter plot, and regression line for -If
Wireline density versus core-plug porosity scatter plot and regression line
Scatter plot and regression line (left) and histogram of probe-based
variability (right) of Rannach formation core plugs
Probe permeability variation observed as a function of visually assessed
rock type (left) and a probe-plug permeability scatter plot (right) for plugs
with limited variability
Schematic illustration of correlation types for porosity and permeability
Summary of selected relationship modeling and assessment methods
according to the nature and spacing of the sampled variable265
Schematic and nomenclature of a property Z measured at points along a
line
Schematic of autocorrelogram with modest (left), little (center), and strong
(right) correlation
The autocorrelogram of an ordered card deck
Schematic illustration of a sample semivariogram
Sample semivariogram for data with a linear trend
Sample semivariogram for data with a step change
Example permeability plots and semivariograms based on the series Z(xi)
= 1 + 0.4 sin(nxj/6) + Ei
Semivariograms for graded and nested patterns
Two examples of geologic periodicities in probe permeability data from
the Rannach formation
Semivariograms at various scales in permeability data from the Rannach
formation
Illustration of anisotropy in semivariograms from the Rannach formation
Semivariogram estimators for horizontal (H) and vertical (V) data sets from
the Algerita outcrop
Illustration of spherical semivariogram model
Schematic of variable sampling density for average porosities
Layout of known points (x) and point (o) to be estimated
Results of Kriging constrained to well-test data
Discrete CDF of Z at the point o (Fig. 12-1) contrasted with the CDF of
the Zi from Example 3
Illustration of the effect of adding additional cutoff points to eliminate
"short hopping"
Schematic of random residual addition procedure
241
242
243
244
245
251
254
255
264
267
272
275
276
277
278
278
279
282
283
284
286
288
291
302
306
310
311
315
Statistics for Petroleum Engineers and Geoscientists
xvii
Schematic convergence of simulated annealing
Convergence rate for Example 4
Distribution of solvent during a miscible displacement in the detailed
simulation of the Page Sandstone outcrop
13-2A Simulated solvent distribution through cross section using a spherical
semivariogram
13-2B Solvent distribution through cross section using a fractal semivariogram
13-4
Spatial distribution of stratification types on the northeast wall of the Page
Sandstone panel
Solvent distribution through cross section using two semivariograms
13-5
Sweep efficiency of the base and dual population CS cases
13-6
Typical Rannoch formation sedimentary profile
13-7
Heterogeneity at different scales in the Rannoch formation
13-8
Spatial patterns of permeability in the Rannoch formation from
13-9
representative semivariograms over various length scales
13-10 Coarsening-up profile in the Lochaline shoreface sandstone
13-11 Vertical and horizontal semivariograms for probe flow rate (proportional to
permeability) in the Lochaline mine
13-12 Permeability profile for the San Andres carbonate generated from a
porosity-based predictor
13-13 Spherical model fitted to a vertical sample semivariogram
13-14 Sample semivariograms in two horizontal directions for the San Andres
carbonate
13-15 Actual and simulated production from a waterflood
13-16A Three-dimensional permeability realization of an east Texas field
13-16B Three-dimensional permeability realizations of an east Texas field with
pinchout
13-17 Saturation distribution in a simulated waterflood of the Lochaline
Sandstone
13-18 Two realizations of the spatial distribution of the permeability field for the
San Andres generated by theoretical semivariograms fit to the sample
semivariograms shown in Figs. 13-9 and 13-10
13-19 The various types of flow regimes for a matched-density miscible
displacement
13-20 Generation of a three-dimensional autocorrelated random field using
orthogonal semivariograms ·
13-21 Schematic reservoir model at two hierarchical scales capturing welldetermined geological scales at each· step
13-22 Schematic of a sequential indicator simulation for a binary system
13-23 Schematic of object model of a fluvial reservoir
322
325
12-6
12-7
13-1
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
347
349
350
354
356
357
357
358
PREFACE
Upon seeing this book, the reader's first reaction might be, "Another book on
geostatistics?" This reaction is natural, for there are excellent texts on the subject,
ranging from the seminal works of Matheron (1967) and Journel and Huijbregts (1978)
through the more recent works of Agterberg (1974), Isaaks and Srivastava (1989), Cressie
(1991), Hardy and Beier (1994), and Carr (1995). Some books come complete with
software (Davis, 1973; Deutsch and Joumel, 1992). The more general statistical literature
also containsnume;rous excellent works (e.g., Raid, 1952; Kendall and Stuart, 1977; Box
et al., 1978).
Yet there are several important aspects that are not treated together in the currently
available material. We write for academic engineering audiences, specifically for upperlevel or graduate classes, and for practicing engineers and geoscientists. This means that
most of the material and examples are based on engineering quantities, specifically
involving flow through permeable media and, even more specifically, permeability. Our
experience suggests that, while most engineers are adept at the mathematics of statistics,
they are less adept at understanding its potential and its limitations. Consequently, we
write from the basics of statistics but we cover only those topics that are needed for the
two goals of the text: to exhibit the diagnostic potential of statistics and to introduce the
important features of statistical modeling. The reader must look elsewhere for broad
coverage on probability, hypothesis testing, properties of distributions, and estimation
·theory. Likewise, we will direct the reader to other papers or books that treat issues in
more detail than we think necessary here.
The role of geology is emphasized in our presentation. For many engineers, statistics
is the method of last resort, when no deterministic method can be found to make sense of
the geological complexities. In our experience, however, statistics and geology form a
very powerful partnership that aids geosystem diagnosis and modeling. We aim to show
that the data and the geology often have a story to tell and analysis of one informs us
about the other. When heard, this story will provide information about further sampling
and the model formulation needed to emulate the important features. The alternative to
reconciling the geology and petrophysical properties is to grasp any model on the shelf
and let the statistical realizations cover our ignorance. Unfortunately, this latter approach
only partially meets engineering needs.
About half of the book is devoted to the diagnostic, or "listening" topics. This
includes the usual tools such as histograms and measures of variability, along with some
newer concepts, such as using geology to guide sampling and the role of variable
additivity. The other half is then aimed at methods for model development, once the
important aspects of the geosystem behavior have been detected and quantified. Here, we
xix
XX
Preface
present a variety of modeling methods, including linear regression. We devote two
chapters to linear regression because it is so common and has considerable utility; we
think even experienced users will find a few surprises here. The final chapter centers on
several field modeling studies that range from the highly deterministic to the strongly
random. In all cases, the statistical diagnosis and geology were essential to the choice of
the modeling method.
The term geostatistics was intended to reflect the quantification of geologic principles
and the reconciliation of the disciplines of geology and statistics. Matheron (J ournel and
Huijbregts, 1978, p. v) points out that "geo" was intended to reflect the structure while
"statistics" signified the random aspect of assessing and modeling ore deposits. However,
geostatistics has more commonly come to refer to a small part of statistical analysis and
modeling. We have abandoned the term "geostatistics" in this text in favor of the older,
and more appropriate, usage of Matheron (1967) of regionalized or autocorrelated
variables. This frees us to cover all types of statistics that might be of benefit to the
geopractitioner. Pedagogically, this allows us to show the progression of statistics from
probability through correlation to autocorrelation using the same terminology and
notation.
As a collaborative effort, this book has afforded all the authors many opportunities to
learn from each other and expand their interests. A book written by one or two authors
would have been easier to produce, keeping the logistical problems to a minimum. The
result, however, would have suffered by being either much more geological or statistical.
We hope the reader will also benefit from this interplay of views and agree that "life on
the rocks" is not so painful as it sounds.
No work of this kind succeeds without help from others. We are especially indebted to
Joanna Castillo, Mary Pettengill, Samiha Ragab, and Manmath Panda for their help.
Heidi Epp, Sylvia Romero and Dr. I. H. Silberberg are to be specially commended for
their copy-editing and technical diligence in dealing with the multiple revisions and
random errors that are the way of life in producing camera-ready books.
LWL acknowledges the patience of Carole and the kids, Leslie and Jeffrey, for allowing
him to hog the computer for so long. JLJ would like to express thanks for the patience
and support of his wife, Jane, and his daughters Cathy and Leanne; the latter two have
waited for that tree-house long enough! PWMC expresses thanks to Kate, William,
Jessica and Hugo for their support over the years that led to this contribution. PWMC
and JLJ both wish to acknowledge the support of the Edinburgh Reservoir Description
Group members. JLJ particularly thanks Heriot-Watt University for a sabbatical leave to
work on this book and his colleagues in the Department of Petroleum Engineering for
covering for him during that absence. PWMC would also like to acknowledge Kingston
University for the opportunity for practicing geologists to get an appreciation of
statistical methods -- an opportunity that led, many years later, to his involvement in this
book. His current post is funded by the Elf Geoscience Research Center. DJG expresses
his love and thanks to Janet, the "mom", and Amanda and Holly, the "soccer dudettes",
for their patience during the never-ending crunch times. DJG also thanks Chevron
Petroleum Technology Co. (formerly Chevron Oil Field Research Co;) for support of
reservoir characterization research, and his many friends and colleagues on the
Geostatistics Team for spirited discussions on the broad application of this technology.
ACKNOWLED M NTS
Figure 1-1 courtesy of Vladimir Z. Vulovic and Richard E. Prange/(c) 1987 Discover
Magazine; 1-2 courtesy of J. P. Crutchfield; 1-3 and 6-11 with kind permission of
Elsevier Science NL; 1-4 reprinted by permission of American Association of Petroleum
Geologists; 3-4, 6-5, 6-8 through 6-10, 13-5, 13-15 through 13-16, 13-19 courtesy of
Society of Petroleum Engineers of AIME ; 11-13 courtesy of American Geophysical
Union; 13-2, 13-3, 13-5, and 13-6 courtesy of Geological Society; 13-12 through 13-14,
13-18 copyright courtesy of Academic Press, 1991.
xxi
1
INTRODUCTION
The modern study of reservoir characterization began in the early 1980's, driven by the
realization that deficiencies in advanced oil recovery techniques frequently ha9 their origin
in an inadequate reservoir description. The litany of problems became commonplace:
wells drilled between existing wells did not have the interpolated characteristics, chemicals
injected in wells appeared (when they appeared at all) in unforeseen locations, displacing
agents broke through to producing wells too early, and, above all, oil recovery was
disappointing. Each of these problems can be traced to a lack of understanding of the
nature of the distribution of properties between wells.
It is surprising that inferring the nature of interwell property distributions is not firmly
rooted in petroleum technology, despite the maturity of the discipline. Engineers
generally constructed a too-simple picture of the interwell distribution, and geologists
generally worked on a much larger scale. Consequently, both disciplines were unfamiliar
with working in the interwell region at a level of detail sufficient for making predictions.
Into this gap has come the discipline of geological statistics, the subject of this book.
Geological statistics has many desirable properties. As we shall see, it is extremely
flexible, fairly easy to use, and can be made to assess and explore geological properties
easily. But itflies in the face of many geologic traditions in its reliance on quantitative
descriptions, and its statistical background is unfamiliar to many engineers. This
statement, then, epitomizes the overall objectives of this book: to familiarize you with
the basics of statistics as applied to subsurface problems and to provide, as much as is
possible, a connection between geological phenomena and. statistical description.
Chap. 1 Introduction
2
1·1
THE ROLE OF STATISTICS IN ENGINEERING AND
GEOLOGY
The complexity of natural phenomena forces us to rely on statistics. This reliance brings
most of us, both engineers and geologists, into a realm of definitions and points-of-view
that are unfamiliar and frequently uncomfortable. Therefore, the basic idea of this section
is to lay some general groundwork by explaining why we need statistics.
We take statistics to be the study of summaries of numbers; a statistic, therefore, is
any such summary. There are many summaries: the mean, standard deviation, and
variance, to name only a few. A statistic is quite analogous to an abstract or executive
summary to a written report that exists to convey the main features of a document
without giving all the details. A statistic gives the main feature(s) of a set of numbers
without actually citing the numbers. Unlike abstracts, however, the specific statistic
needed depends on the application. For example, the mean and standard deviation of a card
deck have no relevance compared to the ordering, a measure of which is also a statistic.
In a broader sense, statistics is the process of analyzing and exploring data. This
process leads to the determination of certain summary values (stati~tics) but, more
importantly, it alsoleads to an understanding of how the data relate to the geological
character of the measured rock. This last point is quite important but often overlooked.
Without it, the data are "boiled down" to a few numbers, and we have no idea whether
these numbers are representative or how significant they are. As we shall see below, the
essence of statistics is to infer effects from a set of numbers without knowing why the
numbers are like they are. With exploration, the data can "tell their story," and we are in
a position to hear, understand, and relate it to the geological character. The geology then
allows us to understand how to interpret the statistics and how significantthe results are.
As we will see numerous times in this book, statistical analysis on its own may
produce misleading or nonsensical results. For example, the statistical procedures can
predict a negative porosity at a location. This is often because the statistical analysis and
procedures do not incorporate any information about the geological and physical laws
governing the properties we are assessing and their measurement. It 1s up to the analyst
to understand these laws and incorporate them into the study. Statistics is a powerful
method to augment and inform geological assessment. It is a poor substitute for physics
and geological assessment.
Definitions
Let us begin by considering two extreme cases of determinism and randomness.
Borrowing from systems language, a system is deterministic if it yields the same output
when stimulated several times with the same input. It is random if it yields a different
Statistics for Petroleum Engineers and Geoscientists
3
output (unrelated to each other) from the same input. Stochastic describes a system that is
part deterministic and part random, a hybrid system. The essential feature here is that the
descriptors apply to the system, although, since knowledge about the system is limited,
we frequently use them to apply to the output. Stochastic implies that there is some
unpredictability in the description of a set of numbers or in a statistic; random implies
complete unpredictability.
On this basis, reservoirs can be placed into one of three categories. The first is strictly
deterministic, having a recognizable, correlateable element at the interwell (km) scale with
a well-understood internal architecture. The second is mixed deterministic and random
(i.e., stochastic), having a recognizable, correlateable element at the interwell (km) scale
with a well-understood gross internal architecture but local random variability, noise, or
uncertainty. Finally, there is the purely random, with no readily identifiable geological
control on the distribution of properties in a heterogeneous flow unit. We will see
examples of all three types.
Some types of statistical operations are quite deterministic because repeated application
of the same input to the same problem will give the same estimate. These procedures
include both regression and Kriging. By saying that a variable is stochastic, we are not
implying that there is no determinism in it at all-merely that there is at least some
unpredictability associated with it. In later chapters, we shall produce several sets of
numbers that are all different but that have the same §tatistic; each set is called a
stochastic realization.
Deterministic Versus Stochastic Systems
In general, deterministic predictions are superior to stochastic predictions. After all,
deterministic predictions have no uncertainty and usually contain significant information
about the nature of the system. Consequently, if the system under consideration is
perfectly understood (in our case, if the nature of interwell property distribution is
perfectly known), then determinism is the preferred prediction method. But if we are
honest, we recognize that there rarely is such a thing as a perfectly deterministic systemcertainly no system that can predict permeable media property distributions. In fact,
exactly the opposite is true; a great many properties seem to be random, even when their
underlying physical cause is understood.
Randomness has always been an unsettling concept for physical scientists who are
used to solving precise, well-defined equations for well-defined answers. Einstein himself
said that "God does not play dice" with the universe (quoted by Hoffman, 1972). We
know that all physical measurements entail some random error; nevertheless, only a
cursory look at the real world reveals an incredible amount of unexplained complexity and
Chap. 1 Introduction
4
apparent randomness-randomness that usually exceeds any conceivable measurement
error.
Causes for Randomness
How can there be randomness when nature follows well-established, highly deterministic
physical laws such as the conservation of mass, energy, and momentum? Curiously, this
dilemma, which is the origin of a breach between physical scientists and statisticians, has
only recently been resolved. We try to illustrate this resolution here.
t
00
0
10
Rate of Spin
Figure 1-1.
•
The outcome of a flipped coin. The shaded dark regions
indicate where the coin will come up "heads"; the light where
it will be "tails." From MacKean (1987).
Statistics for Petroleum Engineers and Geoscientists
5
Consider the epitome of a random event: the flipping of a two-sided, unbiased coin.
The rise and fall of the coin are governed by Newton's second law and the rate of spin by
the conservation of angular momentum-both well-defined physical laws. The first
requires as input (initial condition) the initial velocity at which the coin is tossed. This
is equivalent to the height of the toss. The second law requires the initial angular
momentum or the rate of spin. With both the height and rate of spin specified, we can
plot the results of the deterministic solution as in Fig. 1-1, where the shading indicates
regions having a common outcome, heads or tails.
At the origin, the regions are large and it should be easy to predict the outcome of a
flip. But as the height and rate of spin increase, the regions become small and, in fact,
become quite small for values that are likely to be attained for a true coin flip. If we
imagine that there is some small error in the height and/or rate of spin (such uncertainty
always exists), then it is apparent that we can no longer predict the outcome of the flip
and the coin toss is random.
Randomness, then, is the manifestation of extreme sensitivity of the solution of an
equation (or a physical law) to small uncertainties in initial conditions. From this point
of view, then, few systems are truly random but a great many are apparently random.
There are two requirements for this sensitivity to exist: a nonlinear physical law and a
recursive process. Both are present in nature in abundant quantities (e.g., Middleton,
1991). However, it's surprising just how simple an equation can be and still exhibit
apparent randomness.
May's equation (Crutchfield et al., 1986) is one of many that will generate apparent
randomness:
for Xi between 0 and l. Even though this equation is very simple, it exhibits the two
requirements of randomness. The parameter A controls the extent of the nonlinearity. If
we generate various Xi+ 1 starting with xo = 0.5 we find that the behavior of the Xi for i
greater than about 20 depends strongly on the value of A. For small A, the Xi settle down
to one of two values. But, as Fig. 1-2 shows, for large A, Xi can take on several values.
In fact, for A between 3.800 and 3.995, there are so many values that the Xi begin to
take on the character of an experimental data set. Furthermore, an entirely different set of
numbers would result for xo = 0.51 instead of xo = 0.50; therefore, this equation is also
very sensitive to initial conditions.
Remember that this behavior emanates from the completely deterministic May's
equation, from which (and from several like it) we learn the following:
Chap. 1 Introduction
6
1. The sensitivity to initial conditions means we cannot infer the initial conditions
(that is, tell the difference between xo = 0.50 and xo = 0.51) from the data set
alone. Thus, while geologists can classify a given medium with the help of
modem analogues and facies models (e.g., Walker, 1984), they cannot specify the
initial and boundary conditions.
2. The sensitivity also means that it is difficult (if not impossible) to infer the
nature of the physical law (in this case, May's equation) from the data. The latter
is one of the main frustrations of statistics-the inability to associate physical
laws with observations. It also means that the techniques and procedures of
statistics are radically different from traditional science and engineering approaches.
1.0 t
0.8
0.6
X
0.4
0.2
0.0
3.5
3.6
3.7
3.8
3.9
4.0
'A
Figure 1-2.
Bifurcation diagram for May's equation. From
Crutchfield et al. (1986).
The latter point is especially significant for earth scientists. It seems that if we cannot
infer a physical law for sedimentology, how then can we make sedimentologically based
predictions? The answer lies at the heart of statistics. We must take the best and most
Statistics for Petroleum Engineers and Geoscientists
7
complete physical measurements possible, make sure that sampling is adequate, derive the
best statistical estimates from the data, and then use these to make predictions (in some
fashion) that are consistent with the inferred statistics. Each of these points is touched
upon at various places in this book.
1-2
THE
MEASUREMENTS
Statistical analysis commonly begins after data have been collected. Unfortunately, for
two reasons this is often a very late stage to assess data. First, measurements will already
have been made that could be inappropriate and, therefore, represent wasted time and
money. Second, if the analysis indicates further measurements are needed, the
opportunity to take more data may be lost because of changed circumstances (e.g., the
wellbore being cased off or the cote having deteriorated). Therefore, it is more effective to
perform statistical analysis before or while measurements are made.
How can we analyze and calculate statistics before we have numbers to manipulate?
The answer lies largely in the fact that geology is a study of similarities in rock features.
If a formation property is being measured for a particular rock type, similar rock types can
be assessed beforehand to guide the present collection scheme. In that way, we will have
some idea of the magnitude of the property, how variable it may be, and the scale of its
variation. This information can then guide the sampling scheme to make sufficient
measurements at an appropriate scale where the rock property is most variable.
While it may appear obvious, it is important to make more measurements where the
property varies much than where it varies little. This procedure conflicts with common
practice, which is to collect samples at fixed times or spacings. The result of the latter
approach is that estimates of reservoir properties and their variation are poorest where they
vary most.
It is less obvious that the scale of the measurement is important, too, because
variations in rock properties often occur at fairly well-defined scales (see below for further
discussion of this). Any measurement, representing the effective value on the
measurement volume, may not correspond to any of the geological scales. This
mismatch between the scales of measurement and geology often leads to poor assessment
and predictions. The assessments of properties are poor because the measurements are
representing the values and variabilities of multiples of the geological units, not of the
unit properties. Predictions are degraded because knowledge of the geometries of
variations can no longer be incorporated into the process. When information is ignored,
the quality of predictions deteriorates.
There is also often an uncertainty in the measurement values ansmg from
interpretation and sampling bias. Each measurement represents an experiment for a
Chap. 1 Introduction
8
process having a model and a given set of boundary conditiQns. The instrumentation then
interprets the rock response using the model to determine the constitutive property for the
volume of rock sampled. Two examples of volumes and models are well-test and coreplug permeability measurements. The well test has poorly defined boundary conditions
and the predicted permeability depends on solutions to radial flow equations. A core plug
fits in a rubber sleeve, ensuring a no-flow boundary, and the end faces have constant
pressures applied. This simplifies the geometry so that Darcy's law can be invoked to
obtain a permeability. The well test has a large, poorly defined volume of investigation
while the plug has a very specific volume. Any errors in the assumed boundary
conditions or interpretative model will lead to imprecise estimates of the rock properties.
Sampling bias does not necessarily produce imprecise data. It may instead produce data
that do not represent the actual rock property values. For example, the. mechanical
considerations of taking core plugs means that the shalier, more friable regions are not
sampled. Similarly, samples with permeabilities below the limit of the measuring device
will not be well-represented. During coring, some regions may become rubblized, leading
to uncertainties in the locations of recovered rock and their orientations.
Thus, measurements have many failings that may mislead statistical analysis. The
approach advocated in this book emphasizes the exploratory power of statistics in the
knowledge of the geology and measurements. Through examples and discussion, we
illustrate how dependence solely on statistical manipulations may mislead. At first, this
lessens th~ confidence of. the neophyte, but with practice and understanding, we have
found that the models and analyses produced are more robust and consistent with the
geological character of the reservoir.
1-3
THE MEDIUM
The majority of reservoirs that are encountered by petroleum geoscientists and engineers
are sedimentary in origin. Sedimentary rocks comprise clastics (i.e., those composed of
detrital particles, predominantly sand, silt, and clay) and carbonates (i.e., those whose
composition is primarily carbonate, predominantly limestones and dolomites). Both
clastic and carbonate reservoirs have different characteristics that determine the variability
of petrophysical properties.
Variability in Geological Media
Clastic rocks ~e deposited in the subariel and subaqueous environments by a variety of
depositional processes. The accumulation of detrital sediments depends on sediment
transport (Allen, 1985). When the transport rate changes, erosion or deposition occur.
Sediment transport is a periodic phenomenon; however, for all preserved sequences,
Statistics for Petroleum Engineers and Geoscientists
9
deposition prevailed in the long term. Petrophysical properties are controlled at the pore
scale by textural properties (grain size, grain sorting), regardless of depositional
environment (Pettijohn et al., 1987). Texture is controlled by many primary parameters:
the provenance (source of sediments) characteristics, the energy of deposition, climate,
etc., and their rates of change. Secondary phenomena such as compaction and diagenesis
can also modify the petrophysical properties. Nevertheless, the influence of primary
depositional texture usually remains a strong determining factor in the petrophysical
variability in clastic reservoirs.
In carbonate rocks, primary structure is also important. Carbonate sediments are
biogenic or evaporitic in origin. Primary structure can be relatively uniform (e.g.,
pelagic oozes, oolitic grainstones) or extremely variable (coralline framestones).
Carbonates, particularly aragonite and calcite, are relatively unstable in the subsurface.
Pervasive diagenetic phenomena can lead to large-scale change of the pore structure. The
change from calcite to dolomite (dolomitization) leads to a major reduction in matrix
volume and development of (micro)porosity. · Selective dolomitization of different
elements of the carbonate can preserve the primary depositional control. More often the
effective reservoir in carbonates derives entirely from diagenetic modification, through
dissolution (stylolites), leaching (karst), or fracturing. As a result of these phenomena
(often occuring together), carbonates have very complex petrophysical properties and pose
different challenges to the geoscientist, petrophysicist, or engineer.
As we have outlined, the fundamental controls on petrophysical properties in clastic
reservoirs (textural) are generally quite different from those in a carbonate (diagenetic or
tectonic). It is to be expected that different statistical techniques and measures are needed
to address the petrophysical description of the reservoir medium. Other rock types
occuring less commonly as reservoirs, such as fractured igneous rocks and volcanics,
might require different combinations of the statistical techniques. However, the approach
to statistical analysis is similar to that presented in this book: prior to statistical
analysis, attempt to identify the controlling phenomena for the significant petrophysical
parameters.
Structure in Geological Media
We noted in the previous section that the controls on petrophysical properties are different
in clastics and carbonates. These differences manifest themselves in different large-scale
performance of the medium to an engineered project.
In clastic media, the textural variability is arranged into characteristic structures. These
(sedimentary) structures are hierarchical, and the elements are variously described as stratal
elements, genetic units, and architectural elements for various depo-systems (Miall, 1988;
van Wagoner et al., 1990). These elements are essentially repetitive within a single
10
Chap. 1 Introduction
reservoir, characteristic of the depositional process, and their association forms the basis
of sedimentological and environmental interpretations. Distinctive elements are lamina,
beds, and parasequences in the marine envronment at the centimeter, meter, and decameter
scales, respectively (Fig. 1-3). Groups of these elements are generally called sets.
At the larger scale the stacking of these elements (the architecture) controls the
characteristics of reservoir flow units (Hearn et al., 1988; Weber and van Geuns, 1990).
The geometries of elements can be measured in outcrop (e.g., the laminaset elements in
various shallow marine outcrops in Corbett et al., 1994). The repetitive nature of the
elements results from cyclical processes such as the passing of waves, seasons, or longerterm climatic cycles because of orbital variations (House, 1995).
In carbonate media, similar depositional changes can also introduce an element of
repetitive structure. Often, however, the permeability appears to be highly variable, lacks
distinctive structure, and appears random.
The degree of structure and the degree of randomness often require careful measurement
and statistical analysis. Clastic reservoirs comprise a blend of structure (depositionally
controlled) overprinted by a random component (because of natural "noise" or
postdepositional modification). Clastics tend to be dominated by the former, and
carbonates by the latter. This is by no means a rigorous classification, as a very
diagenetically altered clastic might exhibit a random permeability structure, and,
alternatively, a carbonate may exhibit clear structure. The degree of structure, however, is
one that must be assessed to predict the performance of the medium (Jensen et al., 1996).
Assessing Geological Media
There are various methods for assessing the variability and structure of geological media.
Variability can be assessed by measurements at various increasing scales (Fig. 1-4): thinsection, probe or plug permeameter, wireline tool, drill-stem test, and seismic. The
comparison of measurements across the scales is compounded by the various geological
elements that are being assessed, the different measurement conditions, and the
interpretation by various experts and disciplines. In general, the cost of the measurements
increases with scale, as does the volume of investigation. As a result, the variability
decreases but representativeness becomes more of an issue.
Statistics for Petroleum Engineers and Geoscientists
11
Formation-scale (m tolOO's m
__../__.,/~
~.&~
Figure 1-3.
__../__.,/~
..--!.,&~
Lamination, bed, and bedset scales of sedimentary structure with schematic
permeability variations. After Ringrose et al. (1993).
Structure is usually assessed by statistical analysis of series of measurements.
Geometrical and photogeological data are also used to assess structure at outcrop (Martino
and Sanderson, 1993; Corbett et al., 1994). In recent years, these structure assessments
have become increasingly quantitative. This approach has raised awareness of the
techniques and issues addressed in this book.
Chap. 1 Introduction
12
Stratal
Thickness Extent
Time
Measurement
PARASEQUENCE
BEDSET
BED
LAMINASET
LAMINA
Figure 1-4.
1-4
Shallow marine stratal elements and measurement scales. After
van Wagoner et al. (1990). Reprinted by permission.
THE PHYSICS OF FLOW
Much of the discussion in this book is aimed at developing credible petrophysical models
for numerical reservoir simulators. There are excellent texts on petrophysics .and reservoir
simulation (Aziz and Settari, 1979; Peaceman, 1978), so we give only a rudimentary
discussion here of those aspects that will be needed in the ensuing chapters of this book.
a
Simulators solve conservation equations for the amount of chemical species at each
poil)t in a reservoir. In some cases, an energy balance is additionally solved, inwhich
case temperature is a variable to be determined. The conservation equations are relatively
simple differential equations (Lake, 1989), but complexity arises because each component
can exist in more than one phase; the degree of the distribution among the phases as well
as fluid properties themselves are specified functions of pressure. Pressure is itself a
variable because of a requirement on overall conservation of mass. In fact, each phase can
Statistics for Petroleum Engineers and Geoscientists
13
manifest a separate pressure, which leads to additional input in the form of capillary
pressure relations. Some simulators can model 20 or more species in up to four separate
phases; the most common type of simulator is the black oil simulator that consists of
three phases (oleic, gaseous, and aqueous) and three components (oil, gas, and water).
The power of numerical simulation lies in its generality. The previous paragraph
spoke of this generality in the types of fluids and their properties. But there is immense
generality otherwise. Most simulators are three-dimensional, have arbitrary well
placements (both vertical and horizontal), can represent arbitrary shapes, and offer an
exceedingly wide range of operating options. Many simulators, such as dual porosity and
enhanced oil recovery simulators, have physical mechanisms for specialized applications.
There is no limit to the size of the field being simulated except those imposed by the
computer upon which the simulation is run.
The combination of all of these into one computer program leads to substantial
complexity, so much complexity, in fact, that numerous approximations are necessary.
The most important of these is that the original conservation equations must be
discretized in some fashion before they are solved. Discretization involves dividing the
reservoir being modeled into regions-cells or gridblocks-that are usually rectanguloid
and usually of equal size. The number of such cells determines the ultimate size of the
simulation; modern computers are capable of handling upwards of one million cells.
The discretization, though necessary, introduces errors into a simulation. These are of
three types.
Numerical Errors
These arise simply from the approximation of the originally continuous conservation
equations with discrete analogues. Numerical errors can be overcome with specialized
discretization techniques (e.g., finite-elements vs. finite-differences), grid refinement, and
advanced approximation techniques.
Scale-up Errors
Even with modern computers~ the gridblocks in a large simulation are still quite large-as
much as several meters vertically and several 1O's of meters areally. (The term gridblock
really should be grid pancake.) This introduces an additional scale to those discussed in
the previous section, except this scale is artificial, being introduced by the simulation
user. Unfortunately, the typical measurement scales are usually smaller than the
gridblock scale; thus some adjustment of the measurement is necessary. This process,
called scalingup, is the subject of active research.
Chap. 1 Introduction
14
fnput Errors
Each gridblock in a simulation requires several data to initialize and run a simulator. In a
large simulation, nearly all of these are unmeasurable away from wells. The potential for
erroneous values then is quite large. The point of view of this book is that input errors
are the largest of the three; many of the techniques to be discussed are aimed at reducing
these errors.
The conservation equations alone must be agumented by several constitutitlve
relationships. These relationships deserve special mention because they contain
petrophysical quantities whose discussion will occur repeatedly throughout this book.
Porosity is the ratio of the interconnected pore space of a permeable medium to the
total volume. As such, porosity, usually given the symbol ¢!, represents the extent to
which a medium can store fluid. Porosity is often approximately normally distributed,
but it must be between 0 and 1. In the laboratory, porosity can be measured through gasexpansion methods on plugs taken from cores (¢!plug) or inferred from electrical-log
measurements. Each has its own sources of error .
Like all petrophysical properties, porosity depends on the local textural properties of
the medium. If the medium is well-sorted, it depends primarily on packing. As the
sorting becomes poorer, porosity begins to depend on grain size as well as sorting.
The most frequently discussed petrophysical quantity in this book is permeability.
This is defined from a form of Darcy's law:
-k flP
u=--
JlL
(1-1)
where u is the interstitial velocity of a fluid flowing through a one-dimensional medium
of length L, !1P is the pressure difference between the inlet and outlet, J1 is the fluid
viscosity, and k is the permeability. Permeability has units of (length)2, usually in Jlm2
or Darcys (D); conventionally w-12 m2 = 1 Jlm2 = w-3 mD. mD means milliDarcies.
The superficial velocity is u = Q/A¢1, where Q is the volumetric flow rate and A is the
cross-sectional area perpendicular to the flow.
Equation (1-1), in which k is a scalar, is the simplest form of Darcy's law. Simulators
use a more general form:
u=-J.·(VP+pg)
(1-2)
where u is a vector, p the fluid density and VP the vector gradient of pressure. A, in this
equation is the fluid mobility defined as
Statistics for Petroleum Engineers and Geoscientists
15
k
f.l
A-=Permeability in this representation is now a tensor that requires nine scalar
components to represent it in three Cartesian dimensions.
kxx kxy kxz ]
k = [ kyx kyy kyz
kzx kzy kzz
where each of the terms on the right side are scalars. The tensorial representation is
present so that u and its driving force ('VP + pg) need not be colinear. Most simulators
use a diagonal version of the permeability tensor:
k =[
kx 0
0 ky
0
0 ]
0
(1-3)
0 kz
kx, ky are the x- andy- direction permeabilities; kz is usually the vertical permeability
and is frequently expressed as a ratio kvh of the x-direction permeability. kvh is
commonly less than one, usually much less than one in a typical gridblock. Unless
otherwise stated, when we say permeability, we mean the scalar quantities on the right of
Eq. (1-3).
Permeability is among the most important petrophysical properties and· one of the
most difficult to measure. On a very small scale, it can be measured with the probe
permeameter. The most common measurement is on plugs extracted from the subsurface;
on certain types of media, larger pieces of the core may be used-the so-called whole-core
measurements. All of these measurements are quite small-scale. Far larger volumes--of
the order of several cubic meters--of a reservoir may be sampled through well testing.
All techniques are fraught with difficulties. Consequently, we inevitably see
correlations of permeability with other petrophysical quantities. The primary correlant
here is porosity; the reader will see numerous references to log(k) vs. cp throughout this
book. Permeability has also been correlated with resistivity, gamma ray response, and
sonic velocity.
Chap. 1 Introduction
16
Permeability varies much more dramatically within a reservoir than does porosity; it
tends to be log-normally distributed. Permeability depends locally on porosity, and most
specifically on grain size (see Chap. 5). It also depends on sorting and strongly on the
mineral content of the medium (Panda and Lake, 1995).
The version of Darcy's law appropriate for multiphase immiscible flow is
(1-4)
where the terms are analogous to those in Eq. (1-2) except that they refer to a specific
phase j. The mobility term is now
Again the terms k and J.1. are analogous to those in the previous definition of mobility.
The new term krj is the relative permeability of phase j. Although there is some evidence
for the tensorial nature of krj. this book treats it as a scalar function.
Relative permeability is a nonlinear function of the saturation of phase j,Sj. This
relationship depends primarily on the wetting state of the medium and less strongly on
textural propertie~. Under some circumstances, krj also depends on interfacial tension,
fluid viscosity, and velocity. It is difficult to measure; all techniques involve some
variant of conducting floods through core plugs, which raises issues of restoration, in-situ
wettability, experimental artifacts, and data handling.
Relative permeability is among the most important petrophysical quantitites in
describing immiscible flow. Some measure of this importance is seen in the fractional
flow of a phase j. The fractional flow of a phase is the flow rate of the phase divided by
the total flow rate. With suitable manipulations (Lake, 1989), the fractional flow of
water in two-phase oil-water flow is, in the absence of gravity,
fw =
(1
+ J.l.w kro
J.l.o krw
)-1
fw is, therefore, a nonlinear function of kro and krw. which are themselves nonlinear
functions of saturation. This relationship between saturation and flow is perhaps the
most basic representation of multiphase flow there is.
In addition to the ones mentioned above-porosity, permeability, and relative
permeability-there are a host of other petrophysical properties that are important to
specific applications. In miscible displacements, diffusion and dispersivity are important;
Statistics for Petroleum Engineers and Geoscientists
17
in immiscible displacements, capillary pressure plays a strong role in determining
ultimate recovery efficiency. Here again, though, the disparateness of scales manifests
itself; capillary pressure is exceedingly important on the grain-to-grain scale, but this
scale is so small that we represent its ·effects through the residual saturations in the krj
functions. For this reason also, capillary pressure is an excellent characterization tool for
the small scale.
1-5
ESTIMATION AND UNCERTAINTY
The basic motivation for analyzing and exploring data must be to predict properties of the
formation being sampled. This requirement thus necessitates the production of estimates
(statistics) that amount to some guess about the value of properties at unsampled
locations in the reservoir. These estimates are often the sole focus of attention for many
users of statistics.
Because estimates are guesses, they also have an uncertainty associated with them.
This is because each estimate is the product of the analysis of a limited set of data and
information. The limited nature of this information implies that our assessment is
incomplete and that, with further information, we could supply a "better" estimate. Two
aspects of statistical analysis help the user to be aware of and to mitigate this uncertainty.
The first aspect is that, to every estimate we produce, we can also produce an estimate
of its associated uncertainty. This gives us a powerful method for assessing the impact of
the limited nature of our data sets. We will find that many of the statistics used in
reservoir characterization have uncertainties related to two features of the data. The first is
the number of data in the analysis. Clearly, data represent information and, the more data
we have, the less uncertain we expect our estimates to be. The second is the variability
of the property under study. This is a parameter over which we have no control but, with
geological information and experience, we may have some notion ofits magnitude. For
example, the arithmetic average has an accuracy that is related to the ratio
(data variability)/ ..Jnumber of data.
Such expressions are very helpful in
understanding what is causing estimates to vary from the "right value" and what we gain
by increasing the number of samples. Since information (i.e., number of data) is
expensive, we can find how much an improved estimate will cost. In the case of the
arithmetic average, doubling the accuracy requires quadrupling the number of samples.
The second aspect is that we can guide the analysis according to the measurement
physics and geological subdivisions of the reservoir. This affords the ability to
incorporate into the analysis at every possible opportunity the geological and physical
knowledge of the reservoir. The inclusion of such information is less quantifiable,
unfortunately, than the effects of data numbers and variability, but is nonetheless
Chap. 1 Introduction
18
important in producing statistically relevant estimates. As an extreme example, we can
take the average of the temperature and porosity of a rock sample, 20"C and 0.20, to give
a result of 10.1. The mathematics does not know any different but the result is
uninterpretable. We have no confidence in being able to apply the result elsewhere.
Ridiculous? Yes, but it is extremely easy to ignore information and suppose that the
statistical procedures will iron out the inconsistencies. The result will be poor estimates
of quantities that are only weakly related to the reservoir.
Interpretability and accuracy of estimates also helps in another common engineering
function, comparison of estimates obtained from different kinds of measurements.
Comparisons and the comparability of estimates will be discussed in this book, but the
subject can be very complex. Estimates of average permeability from, for example, welltest and core-plug data are often only weakly comparable. For well tests, the statistical
uncertainties of these measurements is only a small part of the total uncertainties, which
include the domain of investigation, well-bore effects, and model uncertainty. Yet, these
latter aspects are not well-quantified and, therefore, cannot be easily translated into an
uncertainty of the estimated permeability. Nonetheless, error bands for estimates can at
least convey the statistical causes of uncertainty. These can then assist careful, geoengineering judgments to compare estimates.
1 m6
SUMMARY
Statistics is a powerful tool that can supply estimates and their uncertainties for a large
number of reservoir properties. The discipline also provides important exploratory
methods to investigate data and allow them to "tell their story." This tool is best used in
concert with the geological and physical information we have about the reservoir.
Measurements are a vital part of statistical analysis. While providing a "window" to
examine system response, they have several shortcomings. These shortcomings should
be understood to gain the most from the data. Statistical analysis should parallel data
collection to guide the choice of sample numbers and measurement type. Geological
information concerning the scale and magnitude of variability can improve the
measurement process;
Sedimentary systems are the primary subject for reservoir characterization. Both
clastic and carbonate systems often exhibit regularities that can be exploited during
measurement and prediction. These features appear at several scales and, therefore,
measurements with differing volumes of investigation will respond differently to the rock.
These scales and the structure thus need to be understood, and the understanding will help
the statistical analysis during both the data-collection and analysis phases.
2
c
BASIC CONCEPTS
Up to this point, we have relied on your intuitive grasp of what probability means to
make the case for the roles for probability in reservoir description. Before we give details
of methods and applications that use probability, however, we must agree on what
probability is. Therefore, we will devote some space to defining probability and
introducing terms used in its definition.
Central to the concept of probability is the notion of an experiment and its result
(outcome).
Experiment .S - The operation of establishing certain conditions that may produce
one of several possible outcomes or results.
Sample Space n- The collection of all possible outcomes of .S.
. Event E - A collection of some of the possible outcomes of .S. E is a subset of n.
These defmitions are very general. In many statistical texts, they are illustrated with
cards or balls and urns. We attempt to make our illustrations consistent with the basic
theme of this book, reservoir description. An experiment .S could be the act of taking a
rock sample from a reservoir or aquifer and measuring its porosity. The sample space
is the collection of all porosities of such specimens in the reservoir. Figure 2-1
illustrates that the eventE may be the value of porosity (cjJ) that is actually observed in a
core plug.
n
19
20
Chap. 2 Basic Concepts
(L[)
Event
Figure 2-1.
Sketch of an experiment, measuring the porosity ( ¢) of a coreplug rock sample.
Some books call the sample space the population. An individual outcome (a single
porosity) is an elementary event, sometime just called an element. The null set, 0 (no
outcomes), may also be regarded as an event.
An experiment is always conducted to determine if a particular result is realized. If the
desired result is obtained, then it has occurred.
Occurrence of E - An event E has occurred if the outcome of an experiment belongs
toE.
Example 1 a - Defining Events. Suppose there are two wells with the
measured core-plug porosities in Table 2-1.
This reservoir has had 14 experiments performed on it. If we take the event
E to be all porosities between 0.20 and 0.25, then the event has occurred four
times in Weill and four times in Well2. We denote the event as E = [0.20,
0.25]. Square brackets []represent intervals that include the extremes.
The frequency of these occurrences is important because below we will
associate probability with frequency. The event must be well-defined before
Statistics for Petroleum Engineers and Geoscientists
21
Table 2-1. Core-plug porosities for Example L
Welll
0.19
0.15
0.21
0.17
0.24
0.22
0.23
0.17
We112
0.25
0.20
0.26
0.23
0.19
0.21
the frequencies can be calculated. Slight changes in the event definition can
greatly alter its frequency. To see this, recalculate the frequencies in the
above data when the event definition has been changed to beE= [0.19, 0.25].
2-1
COMBINATIONS OF EVENTS
Consider the case of two events, E1 and E2. Both represent some outcomes of the
experiment 5. We can combine these events in two important ways to produce new
events. The first combination includes all the elements of E1 and E2 and is called the
union of E1 and E2.
Union of Two Events- E1 u E2 is the set of outcomes that belong to either E1 or
E2.
The second combination involves only the elements common to E1 and E2. This is
called the intersection of E1 and E2.
Intersection of Two Events· E1 n E2 is the set of outcomes that belong to both E1
andE2.
Both union and intersection are themselves events. Thus, the union of two events E1
and E2 occurs whenever E1 or E2 occurs. The intersection of two events E1 and E2
occurs when bo.th E1 and E2 occur.
Example 1 b- Illustrating Union and Intersection. Referring to Table 2-1 of
core-plug porosities, if E1 = [0.15, 0.19] and E2 = [0.18, 0.24], then
E1 u E2 = [0.15, 0.24] contains eight elements from Well 1 and four
elements from Well2. E1 n E2 = [0.18, 0.19] contains one element from
each well.
Chap. 2 Basic Concepts
22
Of course, both definitions can be extended to an indefinite number of events.
Example 2 -Facies Detection by Combining Events. Suppose that core
description and wireline log analysis produced the diagram in Fig. 2-2 for a
reservoir. We can label events associated with the three lithofacies through
their gamma ray and porosity values: E1 = [GR3, GR4], E2 = [GR1. GR2],
E3 = [4'1. 4>:2], and E4 = [4'3, 4'4]. The three facies can be distinctly
identified using both GR and 4> data: E1 n E3 =Facies 3, E2 n E3 =Facies
1, and E2 n E4 = Facies 2. If only Facies 2 and 3 are present in portions of
the reservoir, then either measurement would be sufficient to discriminate
these facies since E1 u E3 =Facies 3 and E2 u E4 =Facies 2.
Facies 3
GR4
~
0..
<
El
~
GR3
~
ro
~
Cj
GR 2
E2
GR1
Facies 2
Facies 1
8
<1>1
<1>2
E3
Porosity
Figure 2-2.
Hypothetical wireline measurement patterns for three reservoir
facies. Gamma ray response in API units.
The notions of intersection and union lead to one highly useful concept, mutual
exclusivity, and a rather trivial one, exhaustive events.
Mutually exclusive events· Events E1 and E2 are ri:mtually exclusive if both E1 and
E2 cannot occur simultaneously; that is, if E1 n E2 = 0.
Exhaustive sequence of events- The sequence E1, E2, ... ,En accounts for all possible
outcomes such that E1 u E2 u ... u En= Q.
Statistics forPetroleum Engineers and Geoscientists
23
For mutually exclusive events, the intersection contains no outcomes. For example,
E1 n E4 in Fig. 2-2 is mutually exclusive, since the intersection of these two events has
no outcomes (i.e., no lithofacies identified) and can only be the null event 0. If all the
lithofacies in the reservoir have been identified, E1 n E4 = 0. If, during the analysis of
well data, there were portions of the reservoir satisfying both E1 and E4, then there may
be a previously unrecognized facies present or the measurements may be in error (e.g.,
poor borehole conditions).
·
In an exhaustive sequence, one of the constituitive events must occur. The term
exhaustive sequence refers to the sample space and not the experiment. This is the reason
that it is trivial; if we knew the exhaustive sequence of events of all.the porosities in the
reservoir (the sample space), there. would be no need of estimation or statistics, and the
reservoir would be completely described.
Complementary event 3- The subset of Q that contains all the outcomes that do not
belong to E.
The complementary event for E1 in Example 2 is all of the gamma ray values below
GR3 or above GR4, i.e., 31 = (0, GR3) u (GR4, oo). This definition also applies to the
sample space. the parentheses ( ) indicate intervals that do not contain the extremes.
2-2
PROBABILITY
We are now equipped to consider a working definition of probability. It also involves the
connection between statistics and probability.
Probability- Let E be an event associated with an experimentS. PerformS n times
and let m denote the number of times E occurs in these n trials. The ratio m/n is
called the relative frequency of occurrence of E. If, as n becomes large, m/n
approaches a limiting value, then we set p = lim m/n. The quantity p is called the
probability of the event E, or Prob(E) = p. n~oo
The relative frequency of occurrence of the event Eisa statistic of the experimentS.
The ratio m/n is obtained by conducting n experiments. Also associated with the event E
is the notion that, as n approaches infinity, the statistic m/n will approach a certain
value, p. The statistic m/n only tells us about things that have happened, whereas the
probability allows us to make predictions about future outcomes (i.e., other times and/or
other places). The link between the two occurs when we say "we know what m/n is, but
we will never know what p is, so let us assume that p is about equal to m/n." In effect,
we assume that future performance replicates past experience while recognizing that, as n
increases, the statistic m/n may still vary somewhat.
Chap. 2 Basic Concepts
24
Example 3a - Frequencies and Probabilities. Table 2-2 lists the grain
densities (pg) measured from 17 core plugs.
Table 2-2. Core-plug grain densities for Example 3.
Plug no.
1
2
3
4
5
2.68
·2.68
6
7
8
9
2.68
2.69
2.69
2.70
2.69
2.70
2.68
Plug no.
10
11
12
13
14
15
16
17
2.69
2.70
2.74
2.69
2.68
2.68
2.68
2.70
Estimate the relative frequencies of these events:
1. E1: 2.65 ~ Pg ~ 2.68;
2. E2: 2.68 < Pg ~ 2.71;
3. E3: 2.65 ~ Pg ~ 2.71;
4. E4: 0 ~ Pg < oo; and
5. Es: -oo < Pg < 0.
First, the experiment .S consists of taking a core plug and measuring its
grain density. So, in this case, 17 experiments (n = 17) have been performed
and their outcomes listed. The sample space, Q, consists of all possible
values of grain density. We now consider the relative frequencies of these
events.
1. E1 = [2.65, 2.68] occurred for plugs 1, 2, 3, 9, 14, 15, and 16. m =7
so Prob(E1) = 7/17 = 0.412;
2. E2 = [2.68, 2.71] occurred for plugs 4, 5, 6, 7, 8, 10, 11, 13, and 17.
m = 9 so Prob(E2) =9/17 =0.529;
3. E3 = [2.65, 2.71] occurred for plugs 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ll, 13,
14, 15, 16, and 17. m = 16 so Prob(£3) = 16/17 =0.941;
4. E4 =[0, oo) occurred for all the plugs. m = 17 and Prob(£4) = 17/17 =
1.0; and
5. Es =(-oo, 0] occurred for none of the plugs. m =0 and Prob(Es) =0/17 =0.
While we have labeled these frequencies as probabilities, this may not be
accurate. All five of these frequencies are computed from the data obtained by
Statistics for Petroleum Engineers and Geoscientists
25
conducting only 17 experiments. How accurate are they? That is, assuming
that the experimental procedure for grain-density determinations is accurate,
how representative are these frequencies of the whole reservoir? If we
obtained more data, would the ratio 1r1fn change for any of the listed events?
For example, observing that other names for E4 andEs are Q and 0,
respectively, we would claim that Prob(£4) = 1 and Prob(Es) =0 are exactly
the same values as their relative frequencies. This observation, however,
requires knowledge that is not based solely upon the data set.
The probability definition encompasses what we believe to be true about any kind of
modelling. If we can quantify and understand the performance of any set of observations
(here these are the relative frequencies), then we assume that we can estimate, to some
extent, the future performance.
Assessing probabilities is relatively simple by most modelling standards because the
observations do not need to be interpreted, just recorded. This is both the bane and the
strength of statistical analysis. It is a weakness because all subsequent analyses· will be
largely unrelated to a physical cause. For example, we will not be able to say what
caused the grain densities in Example 3; statistics does not distinguish between
Si02 (pg = 2.65 g/cm 3) and CaC03 (Pg = 2.71 g/cm3). It is a strength, however,
when we realize that a very large number of observations, especially those of geological
origin, simply defy quantitative analysis because of their complexity (Chap. 1).
The association of probabilities with frequencies is by no means agreed upon even
among statisticians. This is because it is entirely possible for future events to take a
direction different from that in the past, particularly if there is an outside stimulus. To
proceed, of course, we must accept the hypothesis given by the definition. However,
with further data, it can be tested to determine if the hypothesis still holds. Chapters 5
and 7 treat this topic further.
2-3
PROBABILITY LAWS
The above definitions lead directly to probability axioms or laws. In most cases, they are
rather obvious, as in the following:
Axiom 1: For every event E, 0 ~ Prob(E)
Axiom 2: For the special case E
~
1
= Q, Prob(E) = 1
Axiom 1 says that the probability of any event must be between 0 (no chance of
happening) and 1 (certainty of happening). Axiom 2 says that some event must occur
from the experiment.
Chap. 2 Basic Concepts
26
The addition laws involve mutual exclusivity.
Axiom 3 {Addition Law): Let£1 andE2 be two events, then
This is sometimes known as the fundamental law of addition. It simply says that the
probability of E 1 or E2 is the sum of the individual probabilities less the probability of
E1 and E2. We use the law most in the following special cases.
Axiom 3': If E1 and E2 are two mutually exclusive events (i.e., E1
r1
E2 = 0), then
Axiom 3": If E1, E2, .. ., Ei,··· is a mutually exclusive sequence of events, then
Prob (E1 u E2 uE3 ... u Eoo) = I, Prob(Ei)
i=l
Axiom 3" is simply a generalization of Axiom 3'.
Example 3b- Illustrating the Basic Axioms. Let's reconsider the grain
density events E1, E2, and £3 defined in Example 3a. Recall that
E1 = [2.65, 2.68], E2 = (2.68, 2.71], and £3 = [2.65, 2.71]. Et and E2 are
mutually exclusive events since there is no value of Pg that will satisfy both
E1 and Ez (i.e., E1 r1 Ez = 0). E3 is the combination of both the events E1
and Ez. From the data, we obtained Prob(El) = 7/17 and Prob(E2) = 9/17.
If we apply Axiom 3', we must have Prob(£1 u E2) = 16/17, which is precisely the same value we obtained for Prob(£3).
We note also that the null set may be written as the complement of the sample space,
since
Prob(0) = 1 - Prob(.Q) = 0
The null set is often called the impossible event.
Statistics for Petroleum Engineers and Geoscientists
2-4
27
CONDITIONAl PROBABIUTIES
When Q contains a finite number of outcomes, calculation of event probabilities is
straightforward, if tedious. It is easiest for cases involving a finite number of equally
likely outcomes, such as throwing dice, flipping coins, or drawing balls from the
proverbial urn or cards from a deck. Even such cases require care in conducting the
experiment, however, to avoid influencing the outcomes of later experiments. We first
consider a simple card example.
Example 4a - Conditional Probabilities. Calculate the probability of
selecting two spades in two draws from a normal 52-card deck. Each card is
returned to the deck and the deck thoroughly shuffled before the next draw.
The sample space consists of all pairs of cards: Q = {(A"l', A'¥), (A'¥, 2"l'),
... , (A", K '¥), (A", A4-), ... ) . Since each draw could produce any one of 52
cards, the total number of outcomes is N = 52• 52 = 2704. Remember that
each card could be any one of 52 and that 2704 is just the number of ways to
take 52 things, two at a time (with replacement).
The event E is the drawing of two spades and the number of outcomes
favorable to the event is 13•13 = 169, since there are 13 spades in the deck
and the second draw is with replacement. Therefore the probability of E is
169/2704 = 0.0625, a surprisingly small number.
The experiment is more complicated when the card is not replaced after each
draw. This is because withholding the first card alters the probability in the
second draw. Such changes lead to the notion of conditional probability.
For two events, £1 and Ez, the conditional probability of Ez given that £1
has occurred is defined to be
Prob(Et n Ez)
Prob(£1)
provided that the Prob(E 1) > 0. (The vertical bar between E 1 and Ez on the
left of this definition means "given that.") Actually, we would not be interested in the probability of Ez given that E 1 never occurs because
Prob(Ez I E1) would not make sense. If E1 and E2 are mutually exclusive,
then if E1 has occurred, E2 cannot occur and we have Prob(Ezl E1) = 0.
Usually, we use the above definition to estimate Prob(£1 n E2) as is illustrated in the
next example.
28
Chap. 2 Basic Concepts
Example 4b- Conditional Probabilities Revisited. We repeat the task
considered in Example 4a of calculating the probability of selecting two
spades in two draws from a normal 52-card deck. This time, however, the
first card is held out before the second draw.
The sample space now consists of fewer pairs of cards: Q ={(A"', 2¥), ... ,
(A¥, Kif), (A¥, Ae\!o), ... }. Pairs with identical cards, such as (A¥, A¥)
and (3 3 *), are no longer possible.
*,
We also define the events differently from Example 4a. Let E1 = {drawing a
spade on the first card} and Ez I E 1 = {drawing a spade on the second card
given that the first card is a spade}. What we seek is Prob(Ez n E1). From
the definition of probability,
Prob(E1) = 13/52 = l/4 because there are 13 spades in 52 cards,
and
Prob(Ez I E1)
= 12/51 = 4/17 because there are now 12 spades in 51 cards.
The second probability differs from the probability of drawing a spade in
Example 4a (Ez = 13/52) because the lack of replacement changes both the
number of cards in the deck and the spades available to be chosen.
Rearranging the definition of conditional probability, we have
or a slight decrease in probability caused by the lack of replacement.
The definition of conditional probability can be easily generalized to provide a
multiplication rule for the intersection of several dependent events. Let A, B, and C be
three events. In terms of conditional probabilities, the probability of all three events
happening is
Prob(A n B n C) = Prob(A I BC) Prob(B I C) Prob(C)
where BC indicates the occurrence of B and C. The reader should compare this to the
addition law for the union of events above, where sums, rather than products, are
involved.
Example 5 - Shale Detection in a Core. A probe permeameter is set to take a
measurement every 3 em along a 10 em section of core. The formation
Statistics for Petroleum Engineers and Geoscientists
29
consists of 1-cm shale and sand layers interspersed at random, with 80% sand
in total (i.e., a net-to-gross ratio of 0.8). What is the probability that only
sand will be measured? Assume the probe falls either on a sand or shale, not
a boundary.
Because of the 3-cm spacing, only three measurements are made over the
interval. Define the following events and their respective probabilities:
C =probe measures a layer of sand, Prob(C) = 8/10;
B =probe measures sand given first layer is sand, Prob(B I C)= 7/9; and
A = probe measures sand given first two layers are sand, Prob(A I BC) = 6/8.
We seek Prob(A n B n C). From the definition of conditional probability,
Prob(A nB n C)= (8/10)(7/9)(6/8) = 0.47
So, with three measurements, there is a one-in-two chance of shale layers
going undetected over a 10-cm sample.
2a5
INDEPENDENCE
We can now define what is meant by independent events.
Independent Events- Two events, E1 and E2, are independent if
This result is known as the multiplication law for independent events. When this is
true, the conditional probability of E 1 is exactly the same as the unconditional
probability of E1. That is, Ez provides no information about E1. and we have from the
above
The two-card draws in Example 4a were independent events because of the replacement
and thorough reshuffle. We now apply the independence law to a geological situation.
Chap. 2 Basic Concepts
30
Example 6- Independence in Facies Sequences. Suppose there are three
different facies, A, B, and C, in a reservoir. In five wells, the sequence in
Table 2-3 is observed.
Table 2-3. Facies sequences for five wells.
Position
Top
Middle
Bottom
Welll
c
Well2
B
Well3
Well4
B
WellS
A
A
A
B
A
B
B
c
c
c
A
c
Let event E1 = {A is the middle facies} and E2 = {Cis the bottom facies}.
Based on all five wells, are E1 and E2 independent? If Weill had not been
drilled, areE1 andE2 independent?
For all five wells, Prob(EI) = 3/5, Prob(E2) = 4/5, and Prob(E1 n E2) =
2/5. Since Prob(E1) Prob(E2) = 3/5•4/5 = 0.48 -:1- Prob(E1 n E2), E1 and E2
are dependent.
Excluding Weill, Prob(E1) = 2/4, Prob(E2) = 1, and Prob(E1 n E2) =2/4.
Prob(El) Prob(E2) = 0.5 = Prob(E1 n E2), so E1 and E2 are independent.
Clearly, the data in Weill are important in assessing the independence of
these two events.
Statistical independence may or may not coincide with geological independence. There
may be a good geological explanation why facies C should be lower in the sequence than
either A orB. In that case, the results of Well 1 are aberrant and indicate, perhaps, an
overturned formation, an incorrect facies assessment, or a different formation.
Example 7 -Probability Laws and Exploration. Exploration drilling
provides a good example of the application of the probability laws and their
combinations.
You are about to embark on a small exploration program consisting of
drilling only four wells. The budget for this is such that at least two of these
wells must be producers for the program to be profitable. Based on previous
experience, you know the probability of success on a single well is 0.1.
However, drilling does not proceed in a vacuum. The probability of success
on a given well doubles if any previous well was a success. Estimate the
probability that at least two wells will become producers.
Statistics for Petroleum Engineers and Geoscientists
The different levels of the tree represent the various wells drilled (e.g., S2
indicates success on the second well and F 4 is a failure on the fourth well),
and each branch is one outcome of the drilling program. The probabilities,
corrected for the success or failure of the previous well, are listed on the
segments connecting the various outcomes. Because of the multiplication
law, the probability of a given outcome (a branch of the tree) is the product
of the individual outcomes; these are tabulated at the end of each branch.
Because each branch is mutually exclusive, the probabilities of having a
given number of successful wells is simply the sum of the probabilities of
the corresponding branches. The probabilities shown are those with two or
more successful wells; their sum is 0.0974. The drilling program has less
than a 10% chance of being successful, probably a cause for re-evaluation of
the whole enterprise.
You can verify that this value is nearly twice what it would be if each well
were an independent event, thus illustrating the benefits of prior knowledge.
It is also possible to evaluate the effect of drilling more wells, or banking on
improved technology by taking the single-well success probability to be
larger than 0.1.
31
Chap. 2 Basic Concepts
32
Example 7 provides some fairly realistic probabilities compared to historic success
rates in exploration. More modem procedures factor in more data along the way (more
seismic traces, more core and log data, reinterpretations, etc.), so that the probability of
success in the next well can be more than doubled. For this reason, it is possible that
the probability of success will increase even after a failed well. On the other hand, a
single failed exploration well may lead to the termination of the entire program.
Typically, an exploration lead becomes a prospect when the overall probability of success
approaches 50%.
2-6
BAYES' THEOREM
We now take the idea of conditional probabilities a step farther. From the definition of
conditional probability,
Prob(E1I E2) Prob(E2) = Prob(E1 n E2)
and
Prob(E2I E1) Prob(E1) = Prob(E1 n E2)
The right sides of these expressions are equal, so
Prob(E1 IE2) Prob(E2) = Prob(E2 IE1) Prob(E1)
We can solve for one conditional probability in terms the other:
~
b( I ) . Prob(E2 I E1) Prob(E1)
Prob(E2)
Pro E 1 E2 =
This relationship is called Bayes' Theorem after the 18th century English mathematiCian
and clergyman, Thomas Bayes. It can be extended to I events to give
Prob(Ei IE)= Prob(E IEi) Prob(Ei)
I
I, Prob(E IEi)Prob(Ei)
i=l
where E is any event associated with the (mutually exclusive and exhaustive) events E1.
E2, .. . ,EJ. It provides a way of incorporating previous experience into probability
assessments for a current situation.
Statistics for Petroleum Engineers and Geoscientists
33
Example 8- Geologic Information and Bayes' Theorem. Consider a problem
where we have some geological information from an outcrop study and we
want to apply these data to a current prospect. Suppose that we have drilled a
well into a previously unexplored sandstone horizon. From regional studies
and the wireline logs, we think we have hit a channel in a distal fluvial
system, but we do not have enough information to suggest whether the well
is in a channel-fill (low-sinuosity) or meander-loop (high-sinuosity) sand
body. Outcrop studies of a similar setting, however, have produced Table 2-4
of thickness probabilities that could help us make an assessment.
Table 2-4. Outcrop data for Example 8 (based on Cuevas et al., 1993).
Thickness less than
m
1
2
3
4
Low -Sinuosity
Probability
0.26
0.38
0.56
0.68
High-Sinuosity
Probability
0.03
0.22
0.60
0.70
If we drill a well and observe x meters of sand, what are the probabilities that
the reservoir is a low- or high-sinuosity channel? The difference could have
implications for reserves and well placement. In outcrop, high-sinuosity
channels were observed to be generally wider and have higher connectivity
than the low-sinuosity variety (Cuevas et al., 1993).
We define E1 = {low sinuosity} andE2 = {high sinuosity}. Before any well
is drilled, we have no way of preferring one over the other, so Prob(El) =
Prob(E2) = 0.50. After one well is drilled, the probabilities will change
according to whether the observed sand thickness is less than l m, 2 m, etc.
Let's suppose x = 2.5 m. This is event E. From Table 2-3, we observe that
x falls between two rows of thickness-less-than entries, x :2:2 and x<3. Since
the event E falls within an interval (2 ::;; x < 3), we can more precisely
estimate the occurrence of E by calculating the interval probability of E as
the difference in interval bound probabilities. In this example, we calculate
conditional interval probabilities as Prob(E I E1) = Prob(E1 at upper bound)Prob(£1 at lower bound). From Table 2-3, Prob(E I £1) = 0.56- 0.38 =
0.18, while Prob(E I E2) = 0.60- 0.22 = 0.38.
Applying Bayes' Theorem withE associated to two mutually exclusive
events E1 and E2, we obtain for the revised probabilities,
Chap. 2 Basic Concepts
34
_________P_ro~b~(E
__I~E~l)~P_r~ob~(~E~l~)________
Prob(E1 IE)= Prob(E I E1) Prob(E1) + Prob(E I E2) Prob(E2)
=
0.18•0.50
= 0.32
0.18•0.50 + 0.38•0.50
and Prob(E2I E)= 1- Prob{E1I E)= 0.68. So the outcrop information has
tipped the balance considerably in favor of the high-sinuosity type channel.
We used the maximum information from the thickness measured in the
previous solution. That is, we knew the thickness was less than 3 m and
more than 2 m. If we only knew the thickness was less than 3 m, the
probabilities would be
_________P_r~o~b~(E__
IE~I~)_•_P_ro~b~(_E~l)~--------­
Prob(Ell E)= Prob(E I E1) • Prob(El) + Prob(E I E2) • Prob(E2)
=
0.56•0.50
= 0.48
0.56•0.50 + 0.60•0.50
and Prob(E2 IE) = 0.52. Thus, the change in probabilities is not nearly so
large because less prior information has been added to the assessment.
In Example 8, the probability values for E1 and E2, before measuring the formation
thickness, are called a priori probabilities. Prob(E1 I E) and Prob(E2 I E) are called a
posteriori probabilities. If there were further information with associated probabilities
available (e.g., transient test data showing minimum sand body width), the a posteriori
probabilities could be amended still further.
Example 8 has a feature that makes it appropriate for Bayes' Theorem: the result of
the experiment (measuring the reservoir thickness) did not uniquely determine which of
the two possible scenarios existed. Now consider the following example, where Bayes'
Theorem may not be suitable.
Example 9- Bayes' Theorem and Precise Quantities. We want to know the
. average reservoir porosity, l/Jo, and some measure of possible variations of
¢o, for the Dead Snake reservoir. From well data, we measure porosity to be
l/Jm = 0.22 for this reservoir. We also have available Table 2-5 of average
porosities for eight reservoirs having a depositional environment and a
diagenetic history similar to those of the Dead Snake. Can we use the
information in Table 2-5 to provide a more appropriate estimate for l/JQ?
35
Statistics for Petroleum Engineers and Geoscientists
Table 2-5. Porosity ranges for Example 9.
Average Reservoir Porosity
0.00 ~ cp < 0.20
0.20 ~ cp < 0.24
0.24 ~ cp < 0.28
0.28 ~ cp < 0.32
0.32 ~ cp < 1.00
Number of Reservoirs
0
2
4
2
0
The answer depends upon how we view c/Jm· If we view c/Jm as being
representative and error-free, then we have to set ifJo = c/Jm = 0.22 and any
other information is irrelevant. On the other hand, we have Table 2-5, which
suggests that cfJo could be higher. That is, cfJo =0.22 might not be
representative for Dead Snake. In order to use the information in Table 2-5,
however, we have to admit to the possibility that our estimate (0.22) may
not be correct and quantify that uncertainty. If we can give some
probabilities of error for c/Jm; then Table 2-5 can help amend the a priori
probabilities by use of Bayes' Theorem.
For example, suppose we determine from seismic data that the well is in a
representative location but the measurement c/Jm is prone to error. From previous
~xperience, that error is Prob(0.20 ~ c/Jm < 0.24 I 0.20 ~ ifJo < 0.24) = 0.7 and
Prob(0.20 ~ c/Jm < 0.24 I 0.24 ~ c/Jo < 0.28) = 0.3. That is, there is a 30% chance
we have c/Jm =0.22 while the reservoir actually has porosity between 0.24 and
0.28. (For simplicity, we assume that the probabilities of other outcomes are
zero.) Using Bayes' Theorem, we can calculate the probabilities for ifJo as follows,
using P for Prob:
Prob(0.20 ~ cfJo < 0.24 I cfJm
=0.22) =
P(0.20::>¢m<0.2410.20::>¢o<0.24)P(0.20::>¢o<0.24)
P(0.20~t/lm<0.2410.20~t/lo<0.24)P(0.20~t/IQ<0.24)+P(0.20~t/lm<0.2410.24$t/IQ<0.28)P(0.24~t/lo<0.28)
=
0.7•0.25
= 0.54
0.7•0.25 + 0.3•0.5
and Prob(0.24 ~ ifJo < 0.28 I c/Jm = 0.22) = 1 - 0.54 = 0.46. Th~re are no
other possible porosities for the Dead Snake that are .both compatible with
the measured cfJm =0.22 and the data in Table 2-5. This table provided the
values for Prob(0.20 ~ c/Jo <0.24) and Prob(0.24 ~ ifJo < 0.28). Thus, while
there is a 54% chance the Dead Snake does have a porosity as low as 0.22,
there is a 46% chance that ifJo ~ 0.24.
36
Chap. 2 Basic Concepts
This problem could be made more realistic with more intervals, but the
principle remains the same. Without an error assessment for t/Jm, Bayes'
Theorem cannot be applied because we are strictly dealing with mutually
exclusive events: the average porosity for a reservoir can only be one value
and we claim to know that value to be 0.22 without error. We can derive
error estimates for f/Jm using methods in Chap. 5 (see, in particular,
Examples 6 and 7).
2·7
SUMMARY
We have now covered several important foundation concepts regarding probability.
Probability is defmed in terms of a large number of experiments performed under identical
conditions; it is the ratio of successful outcomes to the number of trials. Conditional
probability allows the experimental conditions to be varied in stipulated ways. Bayes'
Theorem permits additional information to be incorpor~ted into probabilistic
assessments. All these concepts will be exercised in the chapters to come.
3
UN/VA /ATE
1ST /BUT/
S
A random variable is the link or rule that allows us to assign numbers to events using the
concepts in Chap. 2 by assigning a number-any real number-to each outcome of the
sample space. We call the rule X, each outcome is called ro, and the result of applying
the rule to an outcome is denoted X(ro). More formally, a random variable is defined as
follows (Kendall and Stuart, 1977).
A random variable X(•) is a mapping from the sample space ,Q onto the real line (911)
such that to each element co E ,Q there corresponds a unique real number X(ro) with all
of the associated probabilistic features of co in n.
Sometimes X(ro) is called a stochastic variable. We can illustrate what the definition
means using Fig. 3-1.
The idea this should convey is one of uncertainty, not about the rule X but about the
outcome of the experiment. The random variable incorporates the notion that (1) certain
values will occur more frequently than others, (2) the values may be ordered from smallest
to largest, and (3) although it may take any value in a given range, each value is
associated with its frequency of occurrence through a distribution function.
The value X(ro) associated with each element (ro) may not necessarily have any
relationship to co's intrinsic value. Examples 1 and 2 concern two properties of three
reservoir rock samples.
37
38
Chap. 3 Univariate Distributions
Real Number Line
Figure 3-1.
The random variable X assigns a real number to each possible
outcome co.
Example 1 -Rock Sample Porosities. We measure (with no error) the
porosities of the rock samples and call the measured values cp 1, cf!2, and cf!3.
The sample space (Q) is all values in the range 0 to 1 (Le., Q = .[0, 1]).
Each outcome, cp, has a numerical value corresponding to the porosity of the
rock sample. ·In this case, it makes sense to have X be the identity function.
So the event "the porosity of sample one is co" is assigned the value co. We
can order the values (e.g., cfJ3 ~ cp1 ~ cf!2) and establish that some samples
have a larger fraction of pore volume than others.
Example 1 deals with continuous data. Such variables have a distinct ordering because
they relate to a property that can be "sized." That is, one sample can have a larger or
smaller amount of that property. If the ordering is on a scale that has a well-defined zero
point, where there is none of the property, and the variable can always be meaningfully
added or multiplied, then the variable is on a ratio scale, and it is called a ratiometric
variable. Length falls nicely into this category, along with other extensive properties
Statistics for Petroleum Engineers and Geoscientists
39
such as volume, mass, and energy. For example, the combined volume of two
incompressible systems is the sum of the system volumes. Porosity is a ratio of
volumes so it has a well-defined zero point, but only under conditions where the
associated rock volumes are known can porosities be added meaningfully.
A second type of continuous variable (on an interval scale) is the one that has an
arbitrary zero point or cannot always be meaningfully added. Temperature, time,
position, density, pressure, porosity, permeability, resistivity, gamma ray API, and
spontaneous potential deflection are some examples. Intensive variables are interval
types. Sums, differences, and multiplication/division of such variables may or may not
be meaningful. For example, consider the temperatures of two samples of similar
material. Let T 1 = 3o·c and T2 = 1o·c. so AT= T 1 - T2 = 2o·c. Sample 1 is hotter
than sample 2 by 20"C, but sample 2 may have more energy than sample 1 because
sample 2 may have greater thermal mass than sample 1. Combining the two samples
would not give T = 4o·c. Similarly, let q,1 = 0.10 and~= 0.20 for two rock samples.
If these porosities refer to identical volumes of rock, we can deduce that sample 1 has less
void volume than sample 2. ·In general, if we merge the two samples, the total porosity
cjJy is not equal to q, 1 + cp2 but to some weighted average of· cp1 and cft2: cjJy = ll1cp 1 +
A.2 cfJ2, with A. 1 + A.2 = 1 and ).i ;e:: 0 ..
Example 2 -Rock Sample Depositional Environments. We identify the
depositional environments of three samples: sabkha, dune, and interdune.
The sample space (.Q) consists of all depositional environments, including
lacustrine, aeolian, fluvial,- and shoreface deposits among others. Here, we
choose X to beX(sabkha) = O,X(dune) = 1, andX(interdune) = 2. While we
can order the numbers 0, 1, and 2, as we did in Example 1, the ordering has
no significance and the differences (1- 0) or (2- 1) have no significance.
Furthermore, we may even have trouble differentiating some cases where
sands might have features of two or more environments.
Example 2 deals with categorical or discrete data on a nominal (or "name") scale.
There is no quantitative meaning to the numbers attached to the events (categories); dune
is not twice as much of anything as the interdune and there is no natural zero. Similarly,
ordinal scale data, such as bit wear, hardness, and color lack the proportion information
of continuous data. Ordinal data, however, do have an order to the data because the
variable has a size associated with it; sample A is harder thaq sample B, for example, but
we cannot say that it is twice as hard.
All types of data occur in petroleum problems, and we have to be aware of the ways
that we can use the information they contain. · Ratiometric variables contain the most
information (highest level), and nominal variables contain the least information (lowest
level). A wide variety of statistical methods is available for continuous variables, while a
Chap. 3 Univariate Distributions
40
more restricted range exists for categorical variables. Categorical variables can still give
us vital information about a system, however, so that conditional probabilities may be
useful. Recall, for example, the implications for reserves potential if a fluvial reservoir is
from a low- or high-sinuousity part of the system (Chap. 2). There are many examples
in this text where categorical geological information is incorporated, either explicitly or
implicitly, as a conditional probability to help improve assessments.
Higher-level variables can also be changed into lower-level variables if information is
ignored. For example, indicator variables can be generated from ratiometric or interval
variables by setting cutoff values. An ordinal variable, P, might be defined as
P-{Ok<lmD
1 k ;?: 1 mD
based on the permeability k, an interval variable. P contains less information than k, but
it is still useful: Prob(P = 1) is the formation net-to-gross ratio and P is the net-pay
indicator. Recall that net pay is that portion of the formation thought to have sufficient
permeability that it will contribute to economic production.
For the methods in this book to apply, variables should be one of the four types just
discussed, else they are deterministic. There are other types of variable, however, and care
should be taken to ensure that methods discussed in this book are appropriate for the
variables being considered. Fuzzy variables, for example, are one type that is not suitable
for the methods described here. Fuzzy variables have values that may partly belong to a
set, whereas the variables we use definitely either belong or do not belong to a set. For
probabilities of events to be defined, outcomes of experiments have to be distinct and
recognizable as either having occurred or not. See Klir and Fogler (1988) for further
details.
It would be nice to distinguish by notation between random variables, which have
uncertain values, and deterministic variables. For example, random variables might be
denoted by capital letters while deterministic variables could be denoted by lower case
letters. We shall use this convention for generic variables (e.g., x and X). However, the
common usage of certain symbols for particular reservoir properties, e.g., k for
permeability, R for resistivity, cp for porosity, and T for transmissivity, does not obey
any particular rule of this kind. Therefore, we will not strictly adhere to using notation to
make clear the distinction between random and deterministic variables; we will expect the
reader to understand from the context of the problem which variable is which.
The definitions we have introduced do not rule out dependencies between variables in
different sample spaces or between values within the same sample space. We now
consider how the concept of a random variable provides the essential link between
probabilities and distributions.
Statistics for Petroleum Engineers and Geoscientists
3-1
41
THE CUMULATIVE DISTRIBUTION FUNCTION
The most common of generic distribution types is the cumulative distribution function
(CDF). Given a random variable X, the cumulative distribution function F(x) is defined
as
Cumulative Distribution Function (CDF): F(x)
=Prob( X$; x).
In words, F(x) is the probability of finding a value of a random variable X that is less
.than or equal to x. The argument ofF is x, the bounding value, not X the random
variable. Thus, F says something only about the probability of X being less than a
certain value, but says nothing precisely about what X is. Sometimes we find the CDF
defined as the probability of a random variable greater than or equal to x, but this is just
one minus the definition given above because of Axiom 3 (Chap. 2): pC(x) = 1 - F(x) is
the complement of F.
The CDF uniquely defines all of the probabilistic properties of a random variable.
This might seem to rob X of some of its randomness because X must now conform to a
deterministic function, namely F(x). Remember that the adjective random refers to X, not
F(x). The form of the CDF can range from cases where there are an infinite set of X's, to
a finite set of discreteX's, to where there is only one X. The latter is the degenerate case
where the random variable becomes deterministic.
CDF's have the following properties:
1.
2.
0 $; F(x)::;; 1 for all x since F(x) is a probability.
lim
F(x) = 0
X~--«>
and lim F(x)
X~
-too
= 1.
3.
F(x) is a nondecreasing function: F(x+h) ?.: F(x) for any h ?.: 0 and all x.
4.
F(x) is a continuous function from the right for all x:
where h ~
lim F(x+h) = F(x),
0+
0+ means h approaches 0 through positive values.
h~
An important use of the CDF is that it can be used to find out how often events
within a given range of values will occur (i.e., interval probabilities). Suppose we wish
to investigate the random variable X and its frequency of occurrence between a lower
bound a and an upper bound b (a$;b). If we let the events E 1 = (-oo, a] and E 2 =[a, b],
then clearly E 1 and Ez are mutually exclusive. Axiom 3' (Chap. 2) applies and gives
Chap. 3 Univariate Distributions
42
or
Prob(X~
b)= Prob(X
~a)+
Prob(a
<X~
b)
which becomes, upon rearrangement,
Prob(a
<X~
b)= Prob(X ~b)- Prob(X ~a)= F(b)- F(a)
In words, the probability of a random variable having a value between a and b is given by
the difference between the values of the CDF evaluated at the bounds.
All of statistics exhibits a dichotomy between discrete and continuous mathematics.
The former is far more practical than the latter, but continuous statistics are usually more
theoretically tractable.
Discrete CDF's
These are defined by a function F(x) with a set of jumps of magnitude Pi at points xi for
i = 1, 2, 3, ... ,/ such that
Prob(X =xi) =Pi = F(xi) - lim F(x)
X~Xt
The former statement implies that, if we take any two values a and b (a~ b) such that the
interval (a, b) does not contain any of the jump points xi, then
Prob(a ~X~ b)= F(b)- F(b) = 0
Thus, a random variable X cannot fall within the flat segments of the CDF (Fig. 3-2).
This means that X can take on only values at the jump points. In this case, X is a purely
discrete or categorical random variable. Nominal or ordinal data have this sort of CDF.
If the difference between adjacent xj's (i.e., xi+l -xi) is the same for all i, this
difference is called the class size. If we connect the upper comers of this plot with
straight-line segments, the resulting curve is called a frequency polygon.
Statistics for Petroleum Engineers and Geoscientists
43
1
F(x)
X
0
Figure 3-2.
An example CDF for a discrete random variable.
Continuous CDF's
If we consider a CDF for a random variable that is absolutely continuous and
differentiable (this rules out the frequency polygon) for all x, a continuous CDF F(x) will
be defined mathematically as
X
F(x)
= Jf(u) du
-oo
with
F(oo) =
Jf(u)
du = 1
-oo
A typical shape is sketched in Fig. 3-3.
The random variable X pertaining to this CDF is a purely continuous random variable.
Continuous CDF's lend themselves readily to analysis, but they cannot be developed
directly from experimental observations. What we usually end up doing is fitting a
theoretically smooth curve to the frequency polygon or to the data themselves.
44
Chap. 3 Univariate Distributions
F(x)
X
Figure 3-3.
An example CDF for a continuous random variable.
The function f(u) in the definition of F(x) is the probability distribution function
(PDF) whose properties we consider below. But let us first consider an example of how
CDF's can be used to quantitatively convey geologic information.
Example 3 - Stochastic Shales. Shales can influence several aspects of
reservoir performance, including coning retardation, gravity drainage, and
waterflood sweep. In these cases, it is important to know the locations and
areal extent of shales in a reservoir.
The simplest case is when a shale extends over an entire reservoir interval. It
is observed in every well, and its effect on the reservoir flow can be simply
modeled in a reservoir simulator. Such shales are often called deterministic
shales.
The stochastic shale, on the other hand, might appear only in one or two
wells, and its areal extent between wells is unknown. It may not even be the
same shale that appears in several wells. Its effects on reservoir performance
might be quite small or significant. If a number of such shales are dispersed
throughout the reservoir, their impact can be quite large. To simulate the
effects of such shales, some idea must be obtained about their locations and
sizes. This is where the CDF can help out.
Statistics for Petroleum Engineers and Geoscientists
45
It has been recognized (e.g., Weber, 1982) that shale dimensions vary with
the environment of deposition. In general, the areal extent of shales is greater
in low-energy environments. The exact size of any particular shale will be
determined by a large number of factors governed by the circumstances under
which deposition occurred. With the aid of CDF's such as Fig. 3-4, we can
begin to quantify the observed size-environment relationship. Each curve in
Fig. 3-4 is a CDF and, because the probabilities vary with depositional
environment, these curves are conditional CDF's. If Dis a discrete variable
identifying the environment and X is the shale length, each curve can be
represented as F(x I di) = Prob(X ~xI D = di) where d 1, d2 , .. . , d1 are the
depositional environments.
Such CDF's can be used in a qualitative sense, giving the chances that it is
the same shale that appears at two wells.
For example, suppose the di = deltaic or barrier and the distance between two
vertical wells is 1500 ft. There is observed a single shale in each. In the
absence of other information, there is approximately a 50% chance that shales
observed in these two wells are the same. CDF's can be used in a more
quantitative manner, using the Monte Carlo simulation method (Haldorsen
and Lake, 1982).
Stochastic simulation based on Monte Carlo methods involves using a random number
generator to "scatter" a number of shales within a reservoir simulator model and calculate
the performance. The locations of the shales are picked randomly subject to certain
constraints. The sizes of the shales are chosen from the shale-size CDF. The calculated
performance will, of course, depend on the shale distribution in the model. Consequently,
a number of runs-all with different shale distributions-are usually needed to determine how
variable the performance might be. Each new shale distribution is an equiprobable
realization, one possible outcome of the infinite possible number of realizations
(stochastic experiments). Given computer resource limitations, only a finite number of
realizations are possible. The variability in reservoir performance from these realizations
is claimed to be representative of outcomes from the underlying population. Monte Carlo
simulation will be discussed in more detail later in this chapter and Chap. 12.
Chap. 3 Univariate Distributions
46
deltaic, barrier
marine
0
500
1000
1500
2000
Shale Intercalation Length, ft.
Figure 3-4.
Shale size CDF's based on outcrop data. After Weber (1982).
The lines are curve fits to data. Points shown are from Zeito
(1965).
Producing CDF's From Data
For some random variables, we may know from theory what the CDF is. Some common
distributions will be considered in Chap. 4. There are times, however, when we will not
know what the CDF of a variable is. If we can obtain some sample values of the
variable, we can produce an approximate CDF based on those data. Such a CDF is called
an empirical or a sample CDF.
Empirical CDF's are usually produced for non-nominal variables where there is some
natural ordering to the data. If there is no natural size to the variable, the empirical CDF
might change shape, depending upon the number assigned (i.e., the random variable) to
each category. For variables with ordering, we produce an empirical CDF as follows.
Statistics for Petroleum Engineers and Geoscientists
47
1.
Order the data so that X 1 :-::; X2 :-::;X3 :-::; · · · :-::;XI where I is the number of data.
2.
Assign a probability, Pi• to the event Prob(X :-: ; XJ Here we have to make some
assumptions about the assignment. If X is a continuous variable or is discrete and
the uppermost categories may not have been sampled, Pi= (i- 0.5)/I is usually
adequate. We make the assumption here, without any evidence to the contrary,
that each sample value has an equiprobable chance of occurring. If X is discrete
and all categories have been sampled, Pi= iii .
3. Plot Xi versus Pi and, depending on whether the random variable is continuous or
discrete, either connect the lines with stair-steps (e.g., Fig. 3-2) or a continuous
curve (e.g., Fig. 3-4).
Clearly, this estimated CDF is better-more like the actual CDF-the more data we have.
The first probability assignment formula given above in Step 2 differs from the naive
assumption that Prob(X :-::; X1) = 1/I, Prob(X :-: ; X2) = 2/I, · · ·, Prob(X :-::;XI)= 1. This
is because the second probability assignment formula, Pi= iii, forces Prob(X >XI= 0.
It does not allow for the possibility that, if we are dealing with a continuous variable,
there may be unsampled regions of the reservoir where X> XI . A similar argument
holds if X is discrete with possible unsampled upper categories. Each step in p,pi +1 - Pi•
is still 1/I but all the pj's have been reduced by I/2 so that Prob(X > Xn) = I/2. Clearly,
this is still an approximation without data to justify it. There are other formulas for
assigning probabilities, such as Pi= i/(I +1), which give different values to
Prob(X >XI). If the population CDF is known, the optimal probability assignment can
be computed.
Example 4a -Producing Empirical CDF's. Draw empirical CDF's for the
following data (X) and their logarithm (X*= ln(X)): 900, 591, 381, 450,
430, 1212, 730, 565, 407, 440, 283, 650, 315, 500, 420, 714, and 324.
We first rank the data. One ranking suffices for both variables, X and x*,
since ln(X) is a monotonic function. We then define a probability for each
value (xi= Xi and x; = ln(Xi)), using the formula Pi= Prob(X :-::;xi)=
Prob( Xi :-::; x~) = (i -~)I for the I= 17 points. Table 3-1 shows the
calculations.
Figure 3-5 shows the empirical CDF's. The untransformed CDF (left) is
more curved than the logarithmic CDF (right), which is closer to a straight
48
Chap. 3 Univariate Distributions
line. This is because the data set has mostly moderate but a few large values,
whereas taking the logarithm has evened out the spread of the variable with
probability.
Table 3-1. Data for Example 4a.
No.
Xi
X·*
I
Prob. Pi
No.
X·I
X·*
I
Prob. Pi
1
2
3
4
5
6
7
8
9
283
315
324
381
407
420
430
440
450
5.65
5.75
5.78
5.94
6.01
6.04
6.06
6.09
6.11
0.029
10
O.OJt~
11
0.147
0.206
0.265
0.324
0.382
0.441
0.500
12
13
14
15
16
17
500
565
591
650
714
730
900
1212
6.21
6.34
6.38
6.48
6.57
6.59
6.80
7.10
0.559
0.618
0.676
0.735
0.794
0.853
0.912
0.971
1.00
-
0.75
tl<
0.50
1.00
,....._
*I><
I><
VI
VI
*~ 0.50
'-'
..
,.Q
'-'
0
~
0.75
,.Q
Q
loo
~
0.25
0.00
0.25
0.00
250
500
750
X
Figure 3-5.
1000 1250
5.6
6.0
6.4
x*
Empirical CDF's for the data of Example 4a.
6.8
7.2
Statistics for Petroleum Engineers and Geoscientists
3-2
49
ORDER STATISTICS
Some values of x for the CDF F(x) have special names. xo.so is the value where
x =F- 1(0.50) xo.so and is called the median. xo.25 and xo.75 are the first and third
quartiles, respectively, of the CDF F(x). The difference xo.?5 - x 0.2s is sometimes used
as a measure of the variability of X and is called the interquartile range.
=
A quantile of the CDF F(x) is any value xp such that p-l(p) =xP" If we do not know
the population CDF and only have the empirical CDF, we can only provide estimates of
the actual xp's. These estimated values are termed order statistics. For the data in
Example 4a, xo.so is approximately 450.
P~obability
Distribution Functions
Probability distribution functions (PDF's) are a very common statistical tool. They
represent exactly the same information as contained in the CDF, but it is displayed
differently. The CDF takes a "global" view of X, conveying the probability of X being
less than some stipulated value x, The PDF takes a "local" view and describes how the
probability of occurrence of X changes with x. As with CDF's, there is the distinction
between discrete and continuous properties.
Discrete PDF's
Consider the discrete CDF F(x) discussed previously. For each "jump" i and any small
h > 0,
Prob(xi - h <X~ Xi
If we let h
~
+ h) = F(xi + h) - F(xi - h) =Pi
0, we get Prob(X =xi)= Pi·
Thus the jump at the end of each interval represents the probability that X has the
value xi. The set of numbers (pl, P2• P3· ... )plotted against (xl, x2, x3, ... ) is called the
discrete PDF of the random variable X (Fig. 3-6).
If a horizontal line is drawn through the top of each vertical line to a mid-point
between the neighboring vertical lines, or their extension, on the right (left), the plot is a
bar chart or a histogram.
Chap. 3 Univariate Distributions
50
Figure 3-6.
Example of a PDF for a discrete random variable.
Continuous PDF's
Consider a small interval (x, x + ox) from the x axis of a purely continuous random
number X with CDF F(x). Then we have
x+8x
Prob(x <X~ x + ox)
= F(x + ox) - F(x) = f f(u) du =f(x)
ox
X
This forms the basis for interpretingf(x) as a probability:f(x) represents the frequency of
occurrence of a value of x in the neighborhood of x. The best physical interpretation of
f(x) is as a derivative of F(x), because we see from above thatf(x) = dF(x)/dx. A typical
continuous PDF is shown in Fig. 3-7.
The basic properties of continuous PDF's are
1. f(x)
~
0 for all x since F(x) is nondecreasing
Statistics for Petroleum Engineers and Geoscientists
2.
51
Jf(u) du == 1
b
3.
Jf(u) du.
for any a, b (a::;; b), Prob(a <X$ b)=
a
f(x)
0
Figure 3-7.
X
Example of a PDF for a continuous random variable.
A rather curious offshoot of the last property is that the probability that X takes a
particular value x is zero. If we let b approach a, then we have
a
Prob(X ::::: x) =
Jf(x) dx
:::: 0
a
This is a bit disquieting but entirely consistent with the notion of a random variable.
Recall that we should be able to say nothing about a specific value of a random variable.
Both continuous and discrete PDF's will have peaks and troughs. Those values of x
where a peak occurs are called modes. A PDF with one mode is unimodal; a PDF with
two modes is bimodal, etc.
Chap. 3 Univariate Distributions
52
·Empirical PDF's
Otherwise known as histograms, these PDF's are based on samples. They are a very
common method for assessing data behavior but, as we will see in Example 5, they can
mislead.
Empirical PDF's can be produced for any type of variable. The way we produce an
empirical PDF is as follows.
1.
If the variable is not nominal, order the data so that X1
~
x2
~
X3
~
··· ~
x1
where I is the number of data.
2. For continuous variables, divide the interval x1 - x 1 into convenient intervals.
We call these intervals bins or classes. Usually the bins are all of equal size, Ax,
so the height of each interval is proportional to the probability. If we choose too
few bins, the histogram has little character; if we choose too many bins, the
histogram is too bumpy. A rule of thumb is Ax"" 5(X1 - X1)//. For categorical
variables, each bin is one category.
3.
Count the number of data in the ith bin, Ii, and set Prob(xi ~X< xi+ L1x) =Pi=
IiI!for I = 1 to I -1.
4.
Plot Pi versus
xi~
X< xi +L1x for I= 1 to I-1.
As with the empirical CDF, the empirical PDF more closely approaches the population
PDF as I increases. If unequal-sized classes are used, the height of each interval has to be
determined so that its area (not just the height) is proportional to the probability. This is
because the area of the histogram must be unity.
Example 4b- Drawing PDF's. Draw empirical PDF's for the data (X) and
their logarithm (X*= ln(X)) given in Example 4a: 900, 591, 381, 450, 430,
1212, 730, 565,407, 440, 283, 650, 315, 500, 420, 714, 324 ..
By ordering the data, we find thatX1 = 283,ln(X1) = 5.65,X17 = 1212,
ln(X17) = 7.10. The bin size, using the rule of thumb, is about 250 for the
untransformed data while it is 0.40 for the logarithmic data. Table 3-2 shows
the probabilities.
53
Statistics for Petroleum Engineers and Geoscientists
Table 3-2. Data for Example 4b.
No.
1
2
3
4
Freq.
9
6
1
1
Value
250-499
500-749
750-999
1000-1249
Pi
No.
0.53
0.35
0.06
0.06
1
2
3
4
Value
5.60-5.99
6.00-6.39
6.40-6.79
6.80-7.19
Freq.
4
8
3
2
Pi
0.23
0.47
0.18
0.12
The histograms are shown in Fig. 3-8.
-
,...._
0.50 +
*ll<! 0.40 -
0.60-
*ll<!
ll<!
<I
+
ll<!
v
~
VI
ll<!
,t::J
0
;..
~
0.500.40-
v
0.30-
0.30
-
VI
0.20
-
*
0.10
-
*><
0.20-
'-'
I><:
'-'
,.Q
0.100.00
250
0.60
<I
I
I
I
I
I
500
750
1000
1250
X
Figure 3-8.
I
0
;..
~
0.00
5.6
I
I
I
6.0
6.4
6.8
7.2
x*
PDF's for the data and log-transformed data of Example 4b.
The untransformed data have a few large values (2 out of 17) whereas the
logarithmic histogram has a more even spread across the range. Both data
sets have only one peak (unimodal PDF's), and the untransformed data have
the more strongly negatively (right) skewed PDF.
For some applications we would like to draw a smooth curve through the empirical
PDF's. In these cases, nonlinear transforms, as the ln in Example 4b, could cause
difficulties since the area under the smooth curve drifts away from one.
Example 5 -Detecting Modes with PDF's. We now experiment with two
empirical PDF's using the same data set. The data are 6.5, 7.5, 10.5, 12.5,
13.5, 20.5, 25.5, 26.5, 27.5, 29.5, and 37.0. We violate the rule of thumb
for bin size and set Lh: = 10. One histogram will begin at x = 0 while the
second will begin at x = 5. Figure 3-9 shows the results.
54
Chap. 3 Univariate Distributions
~
-<]
0.5 0.48DUU
+
~
v
~
VI
~
........,
,Q
0
0.3 0.2 -
I
0.1-
'
••ouooouuoaao
I..
~
;
0
0
i
I
I
10
20
30
I
40
50
X
Figure 3-9.
Example histograms for identical data sets with different class sizes.
Note how the shift in histogram starting value changes the apparent shape of
each PDF (from unimodal to bimodal) and their skewness. Part of the
problem is the small bin size. A bin size of L1x = 15 would give a more
stable histogram, so the shape would not change so drastically when the
starting point was changed. This example shows that even something as
simple as a histogram can be deceptive and care should be exercised when
interpreting them. The problems with empirical PDF's, as illustrated in this
example, mean that they are rather difficult to deal with in practice. For this
reason, the CDF is the more practical tool, even though the PDF is
conceptually more familiar and more easily understood.
Why Are CDF's and PDF's Important?
There are several reasons why we want to know the distribution of a reservoir property.
They are:
(1) Modeling. Knowing the CDF/PDF, we can produce models of how the property
varies within the reservoir. ·For example, using stochastic shale models as input to
reservoir simulation permits the presence of shales to be evaluated for their impact upon
reservoir performance. Reserves distributions are often modeled using the CDF's of
Statistics for Petroleum Engineers and Geoscientists
55
several reservoir properties. See Examples 7 and 8 in this chapter and Chap. 12 for more
details.
(2) Estimation. Chapter 5 is largely devoted to estimating certain parameters based on
their CDF/PDF. We can make better use of the available data when we know the
population PDF. An example is when we want to estimate the average formation
permeability. An estimate using the PDF may easily have one-half or less of the
variability of an estimate that is calculated ignoring the PDF.
(3) Diagnosis. We can test data to determine whether they are "good" or "bad" by
comparing the sample PDF with other sample PDF's or some theoretical reference PDF.
PDF's have been used .to assess the accuracy of core measurements (Thomas and Pugh,
1989). Significant changes of facies throughout a reservoir may also be detected by
comparing permeability PDF's from within a well or between different wells. This next
example illustrates this use, along with pointing out some limitations to PDF's.
Example 6- PDF's and Geological Elements- The Lower Brent Group
(Middle Jurassic) is a major North· Sea oil-producing interval and commonly
comprises a thick sand-dominated reservoir, without significant shale breaks
(Corbett and Jensen, 1992b). As a single reservoir group, the shape of its
porosity and permeability histograms (sample PDF's) can be used to help
diagnose the presence of units (e.g., facies) and confirm their geological
identification.
The core porosities for this interval (Fig. 3-10) show a predominantly
unimodal, negatively skewed distribution (i.e., most of the data values lie to
the-high end of the data range) clustering at 27%. A second cluster at 3%
suggests the presence of a second, minor grouping. Core-plug horizontal
permeabilities, in contrast, are strongly positively skewed. A logarithmic
transformation of the permeabilities results in a more symmetrical
distribution that is clearly multimodal, suggesting the presence of several
different groups. That is, each peak represents a substantial number of
permeability data near that value, and the peaks could represent different
geological units.
·
Geologically, there are good reasons for splitting the Lower Brent into two
parts, known as the Etive and Rannoch Formations. These geological
elements (formations) are defined on the basis ofwireline log characteristics
(e.g., gamma ray) and descriptions of cores. They are correlateable at
interwell (km) distances. The Etive is a characteristically medium- to coarsegrained, cross-bedded, stacked channel sandstone.. The Rannoch is a finegrained, highly micaceous, cross-laminated, shoreface sandstone. The grain-
Chap. 3 Univariate Distributions
56
size difference produces the contrasting wermeabilities, even though these
units have similar porosities~
TYPICAL NORlll! SEA.
LOWER BRENT SEQUENCE
Cere. !?tug; cats;
Histo.g;rams:
Two modes
I
3:0!
2000
40:0(),
6000
PERMEABilLiff, miDi
KFY
[ill Sandstone- Coal
mMica
D Mudstone ~ Carbonate
Figure 3-10.
concretions
Geological section and histograms of core porosity and
permeability for the Lower Brent Group.
A plot of permeability versus depth (Fig. 3-11) shows the distinction between
formations. The spatial association of the permeability :is an important
geological factor. Most of the high permeability values are in the Etive and
low values are from the Rannoch. This information is not readily appru-ent
from the PDF's because they ignore the sample locations.
Statistics for Petroleum Engineers and Geoscientists
57
The high-permeability Etive is clearly a separate population from the lowpermeability Rannach. The Etive is characterized by unimodal, slightly
negatively skewed porosity and permeability distributions. The Rannach,
which still has bimodal PDF's, could be further subdivided into carbonatecemented and carbonate-free populations. The low-porosity and lowpermeability intervals are discrete carbonate-cemented concretions (often
known as "daggers"). A stratigraphic breakdown on this basis results in two
flow units with approximately symmetrical porosity and log-permeability
distributions.
The Etive, uncemented Rannach, and cemented Rannach are thus "subpopulations" of the Lower Brent population. The histograms and plots of the
petrophysical data versus depth can be used to distinguish and separate the
geological units. In this example, the permeability data are a more powerful
discriminator than the porosity. This means that, while hydrocarbon reserves
might be similar, the elements will have different flow and recovery
characteristics.
Porosity (%)
Log (k)
0 102030
0
Figure 3-11.
5
10 15
Porosity and permeability histograms for the Etive and
Rannach formations in the Lower Brent.
58
Chap. 3 Univariate Distributions
Within the Rannoch sequence, with the help of a series of very fine-scale
probe"permeameter measurements (Corbett and Jensen, 1992), a further
breakdown of the permeability distribution can be made. Some details of this
analysis will be discussed in Chap. 13.
It is possible to determine the basic elements of geological architecture (the laminae),
which tend. to have unimodal PDF characteristics (Goggin et al., 1988; Corbett and
Jensen, 1993b). Much of the reservoir characterization effort in recent years is driven by
the need to identify and provide petrophysical PDF's for the basic geological elements
(Dreyer et al., 1990; Lewis et al., 1990).
The PDF alone may be limited in the determination of effective flow properties,
however. For example, well tests in a braided fluvial reservoir having a symmetrical logpermeability PDF with an arithmetic average of approximately 500 mD showed the
effective permeability to be approximately 1000 mD in one well and approximately 50
mD in a second (Tom-Rivera et al., 1994). These differences have been explained by the
geological organization of the system into large, flow-dominating channels and small
channel, "matrix"- dominated flow, respectively. The differences in effective permeability
are a function of the spatial distribution of the permeability and cannot be ascertained
from the PDF alone. The petrophysical PDF's are, however, good descriptors and help to
confirm geological assessments. '
Consequently, the PDF and CDF can be powerful devices for making better models
and estimates. For example, if PDF and CDF of porosity and/or permeability are
multimodal, more geologic analysis may be required to further subdivide the medium into
flow units with unimodal petrophysical properties. Carefully defined flow units at early
development stages help earth scientists and engineers optimize and manage production
throughout the life of a reservoir. Empirical PDF's and CDF's are the first step towards
exploring and assessing the geological/flow units that make up a reservoir zone. This
process can also highlight aspects of the reservoir that might otherwise be overlooked.
Transforming CDF's
During the statistical modeling of reservoir properties,,random variables with several
different CDF's may need to be simulated. Random-number generators in calculators and
computers often have only one fixed CDF (e.g., uniform random over [0, 1]), so that a
method is needed to transform the computer random variable to one with the desired CDF.
Let X be the continuous random variable, with CDF Fx(x), produced by the computer
and let Y be the reservoir random variable with desired CDF Fy(y). For the moment, we
will assume that Fx(x) and Fy(y) are known functions; they are invertable by virtue of
the general properties of the CDF. The equation y = rf[Fx(x)] will convert the
59
Statistics for Petroleum Engineers and Geoscientists
computer variable to one with the desired CDF. The reasoning behind this relationship
can best be explained using Fig. 3-12.
The key concept in changing one CDF for another is that the probabilities should be
the same (Fy = Fx). The value of y should correspond to the x with the same
probability. Hence, given an x, Fx(x) = Fy(y) for the suitable y. We then apply the
inverse transformation Fjfoto both sides of the equality to obtain y = Vf [F x(x)]. This
relationship applies also to random variables, since X ::::; x is satisfied when x has the
same value as X.
LOO
-><
X
u...
0.75
l
>.
.........
>-
0.50
F (0.25)
0.25
0.00
u...
------
-1.0
0.0
1.0
X or x
Figure 3-12.
=
2.0
0.25
3.0
0
-2.0
-1.0
0.0
1.0
2.0
Y or y
Transformation from random variable X toY with change in CDF.
Computer random-number generators often have the CDF of Fig. 3-12 (left). There
are also tables of four- or five-digit uniformly distributed random numbers in many
statistics books and mathematical handbooks (e.g., Abramowitz and Stegun, 1965) with
this CDF.
Since the PDF and CDF contain the same information, the PDF's of the supplied and
desired variables could be specified instead of their CDF's. The conversion procedure,
however, requires converting the PDF's to CDF's first.
Example 7- Reservoir Property Distributions. A reservoir property has the
following PDF:
y < 0
0 :::; y :::; 1
y > 1
Three numbers, x = 0.33, 0.54, and 0.89, were generated on a computer with
the PDF
Chap. 3 Univariate Distributions
60
fx(x) ==
{~
X<
0
~X~
0
X>
1
1
Transform the x's to y's.
We first convert the PDF's to CDF's.
y
J/y(u) du
Fy(y) ==
==
0
y < 0
f 2u du == y2
O~y~
y
{
-~
0
1
y > 1
1
and
0
X
Fx(x) ==
X<
0
X
J
fx(u) du ==
{
f 1 du
== x
O~x~
1
0
-00
1
X>
1
These CDF's are shown in Fig. 3-13.
1.00
--
0.75
.....:>.
-
)(
X
u.
>-
0.50
u.
= 0.33
0.25
0.00
-1.0
0.0
1.0
X or x
Figure 3-13.
2.0
0
-1.0
0.0
1.0
2.0
Yor y
Transformation for first random number in Example 7.
X is called a uniformly distributed random variable because its PDF is
constant over the interval [0, 1]. Since Fx(x) == x, Fx(0.33) == 0.33,
Fx(0.54) == 0.54, and Fx(0.89) == 0.89. Fy(y) == y2 so Fy(y) = ..JY. Thus,
Y = 0.57, 0.73, and 0.94, respectively.
Statistics for Petroleum Engineers and Geoscientists
3-3
61
FUNCTIONS OF RANDOM VARIABLES
So far, we have only dealt with "simple" random variables. There are instances, however,
when a random variable is presented as a transformed quantity. For example, permeability
and resistivity are often plotted on a logarithmic scale, so that we are actually seeing a
plot of log(k) or log(R). It is evident that, since k and R are random variables and
logarithm is a monotonic function, log(k) and log(R) are random variables. What is the
PDF of k orR, given that we know the PDF of log(k) or log(R)?
If Y = h(X) is a monotonic function of X and has a known PDF, g(y), then the PDF
of X is given by (Papoulis, 1965)
f(x) = g(y)
11;1
This expression can be rearranged to determine g(y) ifj(x) is known. If Y = h(X) is
not ll11."£111QtOI!Iic, then X has to be broken up into intervals in which h(X) is monotonic.
P!:lpoul.is,(1965, Chap. 5) has the details with a number of examples.
Detetmlliniag the PDF when a variable is a function of two or more random variables,
e.g., Z(X, Y}~, is; quite complicated. For example, if Z =X+ Y, where f(x) and g(y) are
the PDF's of two independent random variables, X and Y, then h(z), the PDF of Z, is
given by
h(z) = Jf(z - y)g(y)dy
= Jf(x)g(z
- x)dx
These integliaills represelilll the convolution of the PDF's of X md Y. Papoulis (1965,
Chap·. 7) pmves· tllciis result and considers a number of e':lfamples. It is clear that
detetmtimll)g tFJ:e PDF o:f a function of two or more random variables is not, in general, a
simple task.. There are Solilile results that are useful whenf(x) an<ill g(y) have certain forms,
e.g., have Gawssiu PDF's, and these will be discussed in Chap. 4.
3-4
THE. MONTE CARLO METHOD
There exists, a powerful numerical technique, the Monte Carlo ~thod, for using random
variables in computer p!!Ograms~. If we know the CDF's of the variables, the method
enables us to examme the effects of randomness upon the predicted outcome of numerical
models. Monte Cad~J requii~res: that we have a model defined that relates the input
variables {e.g .., rese~rvoir p;rope:Fties) to the feature of interest (e.g., oil recovery,
62
Chap. 3 Univariate Distributions
breakthrough time, water cut, or dispersion). The model is not random, only the input
variables are random. The distribution (CDF) of the output quantity and, in particular, its
variability are used to make decisions about economic viability, data acquisition, and
exploitation strategy.
Monte Carlo methods can be quite numerically demanding. If many input variables are
random and they all have large variabilities, a large number of runs or iterations of the
model may be needed to appreciate the range of model responses. For example, an input
numerical model for a waterflood flow simulation could require massive amounts of
computer time if all the input parameters, including porosity, permeability, capillary
pressure, and other properties, are considered to be random variables.
Example 8- Reserves Estimates Using Monte Carlo. In petroleum
economics, by far the most frequent use of Monte Carlo is in reserves
calculations. The stock-tank oil initially in place (STOIIP) is given by
STOIIP = ¢ (1- Sw) Ah
Boi
where Ah is the net reservoir volume, B oi is the initial oil formation volume
factor, Sw is the interstitial water saturation, and ¢is the porosity. In this
case, this equation is the model and we are interested in the STOIIP CDF (or
PDF) as ¢, Sw, and Ah may all be considered as independent random
variables. In particular,¢ and Sw can have meaning as independent random
variables only if they represent average values over a given net reservoir
volume, Ah.
A random-number generator in the computer generates random numbers for all
the variables in the STOIIP equation for which the user specifies a CDF or
PDF (Fig. 3-14). For each value of¢, Ah., and Sw, the STOIIP is
computed. This process is repeated several hundred times. The output is a
series of STOUP values that, using the empirical CDF procedure described
earlier, can give the STOIIP CDF. From that CDF, summary statistics such
as the average or median can be calculated.
To use the Monte Carlo method, the distributions for all the input variables
have to be determined. Experience from other fields, data from the field under
study, and geological knowledge all contribute to the selection of the CDF for
each variable. Interdependence between the variables (e.g., low¢ and high
Sw) can be accommodated if it is known how the variables are interrelated.
63
Statistics for Petroleum Engineers and Geoscientists
"$"Generator
~·)b
"STOIIP" Generator
~F(;)olfl~ ·
One realization
of each random
variable
"Ah" Generator
~Ah)b
-..........
(__..
Ah
,-"
Sw
f•(1-S,,)•Ah
/L:::===---..;4-t-f-+__ _______j
/ / ,
~105
F(Ah)
0
Monte Carlo (rnany
b:allzations)
f(STOIIP)
~
RN
I
"Sw" Generator
f(Sw)
F(STOIIP){~
(o-EJ
Scenarios
Figure 3-14.
Schematic procedure for producing STOIIP estimates using Monte Carlo.
Chap. 3 Univariate Distributions
64
The use of Monte Carlo results in reserve estimation varies from company to
company. Some govemments require Monte Carlo simulation results to be
reported when a company wants to develop a prospect.
While Monte Carlo may seem like an easy option to theoretical approaches, it should
be recognized that the results are sensitive to the input CDF's and consequently variability
in the input data will affect the results. For example, in reserves estimation, if 1> and Sw
do not vary much while the rock volume Ah varies considerably, the STOUP PDF will
be virtually identical to the Ah PDF. In Chap. 6, we will discuss further the effects of
variabilities of the arguments upon the resultant variation.
3-5
SAMPUNG AND STATISTICS
In closing this chapter, there are two philosophical issues to be considered: measurement
volumes and the role of geology.
Our view of the reservoir is entirely from probability distributions of the random
variable X (Fig. 3-15), representing some property. We imagine that this property, e.g.,
permeability, porosity, or grain size, exists as a random variable in a physical region.
Each point in the region has associated with it several PDF's: one for each random
variable of interest. When we take samples and measure them, the properties are no
longer random at the sampled locations. We have specific values of the reservoir
properties obtained over a certain volume of the region. For example, the volume of a
core plug is about 10 cm3, whereas the volume investigated by a well-logging device
may be 0.05 m3. In any case, our measurements are averages of the random variables
over a given volume and they cannot reflect the point-to-point variation of the properties.
By measuring samples of nonzero volume, our characterization of the reservoir is based
on averages of the random variable, not the variable itself. When we estimate a parameter
at a point, we are estimating the most likely value of the parameter's average. In the
range of scales of the geological variability present in the reservoir, some scales may be
contained within the measurement volume and, thus, not be recognized in the result. For
example, core plugs might contain several laminations and, therefore, may not represent
the lamination-scale variation present.
Statistics for Petroleum Engineers and Geoscientists
65
x measured locations
0 unmeasured locations
Figure 3-15.
Reservoir property distributions at sampled and unsampled locations.
This point of view has two implications. The first is that all physical measurements
are now actually averages, and that when we in turn average these, we are taking averages
of averages. Thus, it is possible to have measurements over different volumes of material
(e.g., core-plug permeability and transient-test permeability). Trying to reconcile
measurements made on two different volumes can be very difficult, especially since the
character of the PDF's and CDF's will change for different volume scales. This is
especially true when comparing the statistics of porosity averages derived from core/log
data with porosity estimates derived from seismic attributes, even when they are samples
over common depth intervals.
The second implication lies in regarding the original variable as being random. Many
geological processes-sedimentation or diagenesis-are very well-known and quite
deterministic; thus, it is counter-intuitive to think that the result of these processes-the
variable under consideration-is random. We are not saying that the process is random,
however, just that the detailed knowledge of the result of the process is random. This is
almost always the case in physics: the fundamental law is quite deterministic but, as
discussed in Chap. 1, the detailed knowledge of the result of the process (or processes) is
imprecise.
4
FEATU
The probability distribution function (PDF) of a random variable tells quite a bit about the
values the variable takes. For one thing, it says how frequently some values occur
relative to other values. We can also extract "summary" information about the behavior
of the random variable to determine, for instance, an "average" value, or just how variable
the values are (e.g., the variance).
This chapter begins with definitions of the basic statistical operators, principally
moments and their functions, and then moves on to a brief exposition of some of the more
common PDF's. The operators are importantbecause they allow description of any
random variable, regardless of its PDF, but mainly because much of the subsequent
development rests on the properties of these operators.
4-1
NONCENTERED MOMENTS
With the probability distribution functionj(x) known for a continuous random variable X,
we can define the ,th moment as
+oo
Jlr =
f xr
f(x)dx
where r is a nonnegative integer. In making this definition, we have presumed that.f(x) is
integrable.
67
68
Chap. 4 Features of Single-Variable Distributions
The Expectation
By far the most important is the first (i.e., r=1) moment,
E(X) =
f xf(x)dx
(4-1)
-00
referred to as the expected value or expectation of X. E(X) is the mean for a continuous
PDF.
A discrete random variable X can take on one of a series of values: X1, X2, ... ,XJ with
corresponding probabilities Pl· P2· ... PI. That is, Pi Prob(X =Xi). The expected value
of X is defmed to be
=
I
E(X)= LXiPi
i=l
~o E(X} ittll_e _S':!Jll. of !h~~P.Q£.£ibl~.Yl1l1J_~~-.Q{X)Y~jghted QYJ!!~i!-~~~pective prob~
of occurrence. It is relat~ to, but not necessarily the same as, the arltlimeiic·average.
The arithmetic average, X, is computed from M samples X1, X2. .... ,XM taken from the
parent population, which has possible values x 1, x2, ... , Xf
1 M
X--'""
X
-M£..Jm
m=l
Comparing the equations for X and E(X), we see that the arithmetic average appears
to equate Pi with 1/M. This is only partly correct, however, since X= Xi may appear
several times among the samples. For example, several samples might all have the value
x6. When this happens, Pi =kiM for the arithmetic average, where k is the number of
times Xi appeared. Because of the sample size and sampling variation, the sample values
Xi may not represent all the/ possible values of the parent population. For example, if
M <I, all possible values of X could not have been sampled. Hence, X and E(X)
may not be equal. However, for a finite sample there is no other alternative than to
regard X as an estimate of E(X).
For discrete variables, E(X) need not be one of the values Xi. For example, we h~ve
all seen newspaper reports stating that the average number of children per household is
2.6, or some such number. While no household has a fraction of a child, the expectedvalue computation does not account for the fact that X is a discrete variable in. this case
and treats children like lengths of cloth.
69
Statistics for Petroleum Engineers and Geoscientists
Since there are two definitions for the expectation, corresponding to continuous and
discrete PDF's, we should probably have two symbols. Conventional usage does not
make this distinction, probably because the properties of the two are essentially identical.
Properties of the Expectation
Most of the properties follow from the properties of the PDF in Chap. 3 (which in tum
followed from the Axioms in Chap. 2), or the properties of a sum or integral.
E(c) = c
Expectation of a constant c:
Expectation of a constant c
plus (minus) a random variable:
E(X ±c) =E(X) ± c
An immediate consequence of this property is that the expectation of a random
variable with zero mean is zero. Let [X-E(X)] be such a variable, then
E[X- E(X)] = E(X)- E[E(X)] = E(X)- E(X)=
0
Such transformations are common in statistical manipulations. Other properties are
Expectation of a constant c
times a random variable:
E(cX) = c E(X)
Expectation of the sum (difference)
of two random variables:
E(X ± Y) = E(X) ± E(Y)
The last two properties can be generalized to multiple random variables and their sums
or differences:
·J- "\:"
E ("\:"
a· X· + '\:'b ·Y
a· E(X z
·) +
E(Y·)
L.tl-..i.JJJ-L.t
-"\:"
Lb.·J
;
i
j
i
j
As intuitive as these formulas seem, the last actually involve some subtlety. When the
expectation deals with more than one random variable, the definition involves the joint
probability distribution function for X andY, f(x, y):
+oo +oo
E(X + Y) =
f f (x + y)f(x,y)dx dy
Chap. 4 Features of Single-Variable Distributions
70
+oo
+oo+oo
= I i(- 00 ,+oo,xf{x,y)dxdy)+ I IYf(x,y)dxdy
-00-00
1
-oo
+oo
Jxf(x)dx+ fyj(y)dy=E(X)+E(Y)
-00
+oo
+oo
where If(x, y)dx
= f(y)
and If(x, y) dy )= f(x) are the marginal PDF's for Y and X,
respectively.
The properties described above carry over to more general functions of random
variables, but the sum/difference property is all that is needed here. Discussion of the
properties ofj{x, y) falls under the subject of bivariate distributions, covered in Chap. 8.
There is, however, a final property that requires a further restriction on the random
variables.
Expectation of the product
of two independent random variables: E(XY)
=E(X)E(Y)
For this to be true we must be able to write the joint PDF as f(x, y) = f(x)g(y). You will
no doubt recognize this property of the PDF and the definition of independence of
random variables from the Multiplication Axiom in Chap. 2.
E(XY) =
+oo
]
_1+oo _1+ooxy f(x) g(y)dx dy = [ _1+ooxf{x)dxjl [_lY
g(y) dy = E(X) E(Y)
We defer further discussion of dependent random variables until Chap. 8.
Statistics for Petroleum Engineers and Geoscientists
4~2
71
CENTERED MOMENTS
The rth centered moment of a random variable is
+=
J.l~ = E([X- E(X)Y} =
f [x- E(X)]' f(x)dx
The expectation inside the integral raises the possibility of several ways to estimate J.l~.
For example, the integral may be approximated as a finite sum and the E(X) from a
continuous PDF. In practice, Vfe will almost always be estimating both expectations from
a finite data set. Of course, J.lr must satisfy all of the properties of E(J() with respect to
the PDF.
Variance
One of the central concepts in all of statistics is the second (r = 2) centered moment,
known as the variance and denoted specifically by Var(X).
+oo
Var(X) =
f [x- E(X)J 2 f(x)dx
-00
Using the properties of the expectation, Var(X) can be written as the difference between
the expectation of x2 and the square of the expectation, of X, E(X)2:
Var(X) = E ([X- E(X)]2} = E(X2)- E(X)2
(4-2)
without loss of generality. (In this formula and elsewhere, we adopt the convention that
E(X)2 means that the expection is squared.) This form is actually more common than the
original definition. The discrete version of Var(X) for I values is
I
Var(X) =
'L [xi - E(X)J 2 Pi
i=l
Compare this with the formula for estimating the variance from M samples X1, X2, ... ,XM:
Chap. 4 Features of Single-Variable Distributions
72
M
s
=~ I
[Xm- E(X)]2
m=l
which requires that the mean of X be known. Observations similar to those about the
discrete expectation also apply to this estimation formula.
The variance is a measure of the dispersion or spread of a random variable about its
mean. Other measures exist, such as the mean deviation, E [IX- E(X)I]. The variance,
however, is the most fundamental such measure and, as we shall see, most other measures
of variability are derived from it.
Properties of the Variance
Many of the variance's properties follow directly from those of the expectation operator.
However, its most important property is that it is nonnegative.
Intrinsic property:
Var(X)
~
0
The reason this is so (the equality holds only in the degenerate case of no variability)
follows directly from the basic definition wherein only squares of differences are used
inside an integral andf(x) ~ 0. Of course, an absolute value or any even r will yield a
nonnegative moment, but these are too complex mathematically. The reason that this
seemingly small observation is so important is that when we minimize the variance in the
following chapters, we can be assured that our search will have a lower bound. The
following example illustrates this.
Example 1 -Minimizing Properties of the Variance. You might ask why the
centering point for the variance is the mean instead of some other point, say
the median. We now show that using the mean will make the variance a
minimal measure of dispersion.
For this exercise only, let us define the variance as
Var(X)
=E[(X- a)2]
where a is some arbitrary constant. To find the minimum in Var(X), take its
derivative with respect to a and set it to zero.
Statistics for Petroleum Engineers and Geoscientists
73
dVar(X)
-too
da
= -2 (x- a)fl.x)dx = 0
f
-oo
noting that the differentiation is not with respect to the integration variable.
But using the definition forE(), we find that the minimum variance occurs
when
E(X) =a
or that the variance, as originally defined, is itself minimized. .
To expose the remainder of the variance's properties, we parallel those of the
expectation.
Variance of a constant c:
Variance of a random variable
plus (minus) a constant c:
Var(c) = 0
Var(X± c)= Var(X)
These properties are consistent with the idea of the variance being a measure of spread,
independent of translations. There is, of course, no spread in a constant.
Variance of a random variable
Var(cX) = E[(cX)2]- E(cX)2
times a constant c:
= c2E(X2) -c2E(X)2
= c2Var(X)
Unlike the analogous property for the expectation, this is not intuitive. However, since
the constant can be negative, it is entirely consistent with the idea of a nonnegative
variance, since Var(cX) :=:: 0.
Variance of the sum (difference)
of two independent random variables:
Var(X±Y) =E[(X±Y)2] -E(X±Y)2
= E[X2 ± 2XY + y2] - E(X)2 ± 2E(X)E(Y) -E(Y)2
= E(X2)- E(X)2 + E(Y2)- (Y )2 = Var(X) + Var(Y)
since E(XY) = E(X)E(Y)
74
Chap. 4 Features of Single-Variable Distributions
The last two properties can be combined into
(4-3)
Var(L aiXi) = 'La[var(Xi)
i
Remember that this property is valid for independent Xi only.
Example 2- Variance of the Sample Mean. We can use Eq. (4-3) to derive a
traditional result that has far-reaching consequences. Consider the drawing of
I samples from a population containing independent Xi. The variance of the
XiisCJ 2 .
The variance of the sample mean
X is
1 I
Var( X)= Var([
Xi)
I
i=l
Since the Xi are independent, we can use Eq. (4-3) directly as
But, since Var(Xi) = CJ2,
I 2 cr2
Var( X) =2 CJ = I
I
(4-4)
Equation (4-4) says that the variance of the sample means decreases as the
number of samples increases. This makes sense in the limit of I approaching
infinity, since the variance must then approach zero, each draw now
containing all of the elements in the population. Of course, the variance of
the mean is CJ2 when I= 1.
Equation (4-4) also suggests that a log-log plot of the square root of the
variance (the standard deviation) of the sample mean versus I will yield a
straight line with slope of -1/2. Such a plot is the precursor of the rescaled
range plot used to detect spatial correlation in geologic data sets.
Statistics for Petroleum Engineers and Geoscientists
75
The procedure used in Example 2-manipulating a sum until the term inside the
summation contains no subscripts-is a common statistical approach. This operation
changes a summation into a multiplication to yield a result that is substantially simpler
than the original equation.
Properties beyond this, for example the product of two random variables, are even
more complicated but are not necessary for this text. We will revisit many of these
properties in the discussion of correlated random variables in Chaps. 8 and 11.
4B3 COMBINATIONS AND OTHER MOMENTS
There are a few other related functions of these moments.
Standard Deviation
The standard deviation is the positive square root of the variance
SD = +.VVar(X)
The standard deviation is a very familiar quantity. While Var(X) has the units of x2, the
SD has units of X, making it more comparable to the mean.
Coefficient of Variation
The coefficient of variation is the standard deviation divided by the mean:
SD
Cv= E(X)
Cv is one of the most attractive measures of variability. It is dimensionless, being
normalized by the mean, and varies between zero and infinity. See Chap. 6 for more
discussion.
Coefficient of Skewness
The coefficient of skewness is the third centered moment divided by the second centered
moment:
76
Chap. 4 Features of Single-Variable Distributions
Y1 is a measure of the skewness of a PDF. y1 < 0 indicates a PDF skewed to the right
(i.e., positively skewed) and Y1 > 0 indicates a PDF skewed to the left (i.e., negatively
skewed). Y1 = 0 indicates a symmetrical PDF.
Coefficient of Kurtosis
The coefficient of kurtosis is the ratio of the fourth to second moments:
r 2 is a measure of the peakedness of a PDF. The normal distribution (y2 = 0) is
mesokurtic. A PDF flatter than the normal (r2 > 0) is platykurtic; one more peaked than
the normal distribution (y2 < 0) is called leptokurtic.
These last three definitions are more for terminology than usage. They are rarely used
and not used at all in the remainder of this text. But the expectation and variance
properties will recur frequently.
Let us now loo)c at a few common PDF's and some of their properties. We will begin
with a discrete variable.
The Binomial Distribution
Consider an experiment with only two possible outcomes: E 1 = a "success" with
Prob(El) = p and E2 = a "failure" with Prob(E2) = l - p. E 1 and E 2 are obviously
mutually exclusive in a single experiment, but several repetitions of the experiment
would produce combinations of successes and failures. We also take the experiments to
be mutually exclusive.
Statistics for Petroleum Engineers and Geoscientists
77
Consider the probability of getting s successes in I trials for a given p. Put another
way, we calculate Prob(R = s) where R = Lxi, i=1, ... ,/,and Prob(Xi = 1) = p and
Prob(Xi = 0) = 1 - p. This sort of problem might arise, for example, when estimating netto-gross ratios for a sand-shale sequence (Xi = 1 for sand and Xi = 0 for shale).
To answer this, we consider all possible ways of getting s successes in I trials with
each combination multiplied by its appropriate probability p.
Prob(R = s) =p
[I
s
=s! (/-s)!
·
ps (1 - p)l-s
(4-5)
The symbol I! is read "I factorial" and it means /(/-1)(/-2)· ··1. This function is available
on scientific calculators. Another way of writing
L!
:J-s)!] is 1cs• the number of
combinations of I things taken s at a time.
Equation (4-5) is the binomial distribution, an important result in statistics. Each term
in the equation has physical significance: the first term is the number of ways in which
there can be exactly s successful outcomes, the second is the aggregate probability of s
successes, and the third is the aggregate probability of (/ - s) failures.
Example 3- Using the Binomial Distribution for Thin-Section Porosity
Estimation. Consider a point count on a vugular carbonate thin-section that
has 50% porosity. The microscope cross hairs will fall either on the matrix
(Xo = 0) or on an opening (X 1 = 1). What is the estimated porosity based on
the number of samples we take?
If I is the total number of samples taken and R the number of times we draw
1/J = 1, then the observed or estimated sample porosity is
-
1/J=
RX1 + (/ -R)Xo
R
I
I
-
Because the thin-section has 50% porosity, Prob(l/i = 1) = 1/2. Hence, the
probability of R points out of I samples falling on vugs is
p = Prob(R
(1)/
= r) = r! (/I!_r)! 2
For I= 2 we can haveR= 0, 1, 2. From the binomial distribution the various
probabilities can be plotted as !!.1 Fig. 4-1 (left). We can replaceR on the
abscissa with the ratio R/1 (= 1/J) to get Fig. 4-1 (right).
Chap. 4 Features of Single-Variable Distributions
78
Figure 4-1.
Probability distribution functions for R and
Prob(f/l= 1) = 0.5.
i
for I = 2 samples and
This is a very simple discrete PDF for (f. Clearly, we could repeat this for
any/. FQ! example, if I= 6 we have Fig. 4-2 (left) for R = 0, 1, 2, 3, 4, 5, 6
and the lfl PDF (Fig. 4-2, right).
PR_
~
0.3
0.3
0.2
0.2
0.1
0.1
01234
Figure 4-2.
56
0
Probability distribution functions for R and
Prob(lfl = 1) = 0.5.
0.5
i
1
for I= 6 samples and
Apart from showing an application of the binomial distribution, this exf!!Uple
illustrates an important issue: the variability of estimated quantities ( cjJ in
this case) that depend upon data. We always estimate quantities based on
some number of samples and those estimates are just that: estimated
quantities that are themselves random variables.
79
Statistics for Petroleum Engineers and Geoscientists
Unless we counted an extremely large number of points, ljJ would always be
subject to some variability. We can get some idea of the variability by
looking at the probabilities of the extreme events Prob ( ljJ = 0) and
Prob (cp = 1). Eor I= 2, Prob (p = 0) == Prob (l/J = 1) = 0.25, whereas when
I= 6, Prob ( q; = 0) = Prob ( q; = 1) = 0.016. By tripling the sample size, the
probabilities of the extremes have dropped considerably.
The mean value of R for I trials is
I
E(R)=
L i Pi pi (1- p)l-i =Ip
i=O
This result follows from substituting Eq. (4-5) and using the identity
I
L JC'i xi= (1 +Xi
i=O
By a similar process, the variance is given by Var(R)
in the following example.
= Ip(l - p).
These results are used
Example 4 -Mixtures of Random Variables in Geological Sediments.
Consider a formation unit consisting of two elements, 1 and 2, which
interfinger in proportions p and (1- p), respectively. If each element has a
random value of permeability, k1 and k2 with means E(k 1) = 111 and
E(k2 ) = 112 and variances V ar(k1 ) =a~ and V ar(k2) =a;, respectively, what
are the mean and variance of samples taken from the combination? Assume
that each measurement consists only of material from one of the two elements
and that k 1 and k2 are independent.
Let k be the sample permeability, so that
where X is a binomially distributed variable. If X = 0, the sample is entirely
from element 2; if X= 1, it is from element 1. Since the proportion p of the
unit consists of element 1, Prob(X = 1) = p. We wantE(k) and Var(k) in
terms of p, Jll, Jl2• and
cr~.
The mean value is
Chap. 4 Features of Single-Variable Distributions
80
E(k)
=E[Xk1 + (1 - X)k2J =E(Xk1) + E[(l - X)k2J
=E(X)E(k1) + E(l- X)E(k2)
because the variables X, k 1, and k2 are all independent. Thus,
E(k) = Pf..ll + (1 - p)f..lz
This result agrees with what we would expect. Similar reasoning gives
Var(k) =E[Xk 1 + (1- X)k2]2 - E[Xk 1 + (1- X)k2]2
=
E(X2)E(~) + 2E(k 1)E(k2)[E(X) - E(X2)] + E(~)
- 2E(k~)E(X) + E(k~)E(X2)- [pf..ll + (1 - p)f..L2]2
The total variability of k arises from three factors: the variability of element
1 multiplied by the amount of 1 present; the variability of 2 also weighted by
the amount present; and the additional variability arising because the mean
permeabilities of the elements are different. The third factor is maximized
whenp= l/2.
Realistic sediments that conform to this example include eolian deposits.
Dune slip face sediments consist of grain fall and grain flow deposits. Each
deposit has different grain size and sorting and, hence, will have a distinctly
different permeability from the other deposit. Probe or plug measurements
may be taken in a number of slip face deposits, and the apparent variability
could be quite different from the variability of either element.
Mixtures of random variables cause the resulting PDF to be a combination of the
component PDF's. Such PDF's are called heterogeneous or compound distributions.
Multimodal PDF's often arise because the measurements were taken in sediments with
distinctly different properties.
Statistics for Petroleum Engineers and Geoscientists
81
The Central Limit Theorem
It takes considerable mathematics (Hald; 1952), but much less intuition, to see where the
binomial PDF is going as I approaches infinity. The discrete PDF above becomes
continuous and the PDF itself approaches a normal distribution, that bell-shaped curve we
all have come to know and love. Actually, the approach to normality is quite close after
only about I= 20 when p = 0.5. It takes considerably more points when pis closer to 0 or
l, but the-normal distribution results as Fig. 4-3 illustrates. You can also see the
progression of the mean and variance of the binomial distribution in Fig. 4-3.
According to a fundamental result of applied probability known as the Central Limit
Theorem (CLT), a random variable X, generated by adding together a large number of
independent random variables, will usually have an approximate normal distribution
irrespective of the PDF's of the component variables. This remarkable theorem says that
a normal distribution will result from the binomial distribution as I approaches infinity
regardless of the value of p. Conversely, when we observe a normal distribution in
practice, we assume these attributes are satisfied.
0.40
n
0.35
PR
=10
0.30
0.25
20
0.20
0.15
0.10
0.05
0.00
0
20
40
60
80
100
R
Figure 4-3.
The progression of a binomial PDF to a normal distribution for p = 0.9.
An example of the CLT is core-plug porosity. Each plug consists of numerous pores
that each contribute a small amount to the plug pore volume. Hence, the total porosity of
plugs is likely to be nearly normally distributed. Indeed, in many clastic reservoirs,
porosity is nearly normally distributed. The binomial distribution is not even necessary
Chap. 4 Features of Single-Variable Distributions
82
for this argument (we used it mainly as an illustration). As long as the underlying events
are independent and additive and there are a large number of them, the PDF will be
normal (Blank, 1989). The CLT is as close as we get to a physical basis for statistics and
it accounts, in part, for the popularity of the normal distribution in analysis. Many of the
simulation techniques and tools (to be discussed in Chap. 12) require normally distributed
random variables.
In light of this result, we can also see how errors resulting from the measurement of a
physical quantity could have a distribution close to normal. Typically, errors arise as a
combined effect of a large number of independent sources of error. These errors,
however, may not perturb the measurement very much, depending on the relative
magnitude of the property and the associated noise. In addition, we would like the
deviations (residuals) of a measurement away from a model to be normal, since this
means that the model accounts for all of the features of the measurements, apart from
unquantifiable influences.
The Normal (Gaussian) Distribution
In addition to the CLT, the normal distribution is hoary with statistical import.
If a random variable X has a normal distribution, its PDF is uniquely determined by
two parameters (j.l and a ) according to
f(x;
0',
J.l) =
.V 2~0'2 exp [- ~ (
~
x .U )
2
].-=
< X < oo
(4-6a)
which graphs as shown in Fig. 4-4.
You can easily show that E(X) = f.1 and Var(X) = a2. We will represent the normal
distribution with the short-hand notation N(mean, variance). So X~ N(3, 9) means X is
normally distributed with a mean value of 3 and a variance of9. The normal CDF is
X
F(x; a, f.l)
=-
1-
.V 2na2
Jexp [-.!2 (
!___::___Y:. )
a
The integral function above is called the probability integral.
2] dt
(4-6b)
83
Statistics for Petroleum Engineers and Geoscientists
Both the normal distribution and the probability integral can be made parameter-free
by redefining the variable as Z = (x - f.L) I a, known as the standardized normal variate.
Thus E(Z) =0 and Var(Z) =1, and we say that Z ~ N(O, 1) so that
z
F(z) =
& f exp(-~ r)
dt
(4-6c)
Extensive tables for N(O,l) are printed in many statistics books; see Abramowitz and
Stegun (1965).
1
F(x) or f(x)
0.8
F(x)
0.6
0.4
0.2
-3
X
Figure 4-4.
The continuous normal PDF and CDF.
The normal distribution occurs anywhere there are observations of processes that
depend on additive random errors. A good example is in diffusion theory, where
assuming a random walk of particles will lead to a concentration distribution having a
normal shape (Lake, 1989, p. 157). In this case, the relevant function is the error
function, defined as
Chap. 4 Features of Single-Variable Distributions
84
0
erf(x) =2-
fe-t
-{;.X .
2dt
which is related to Eq. (4-6c) as
(4-6d)
Probability Paper
How do we know when a variable is normally distributed? There are several fairly
sophisticated means of testing data, but the most direct is to plot them on probability
paper. We can invert the standard normal CDF given in Eq. (4-6d) to
z = -"'J'2 erf- 1[1 - 2F(z)]
(4-6e)
The quantity on the right of this equation is the inverse error function. A plot with the
inverse error function on one axis is a probability plot (Fig. 4-5).
t
X
5
10
20 3040 506070 80
90
95
Probability(%)
Figure 4-5.
Probability coordinates.
99
Statistics for Petroleum Engineers and Geoscientists
85
There is a one-to-one relationship between the value of the normalized variable Z and
the probability that some value less than or equal to Z will occur (i.e., F(z)).
Consequently, we can talk in terms of variable values (z) or their associated probabilities
F(z). In Fig. 4-5, we mark off the horizontal axis in terms ofF, expressed as a percent. If
data from a normal CDF are plotted on the vertical axis, then a straight line will result
with this type of paper. A log-normal distribution will similarly plot as a straight line if
the vertical axis is logarithmic. Either Z or the original X may be plotted, since the
transforms involve simple arithmetic. Plotting X is a commonly accepted way of testing a
set of data for normality and for determining the variability of a set of data.
Probability plotting facilities based on the normal distribution are often found in
statistical packages for computers. Smaller systems, however, may lack this feature. In
order to produce a probability plot, use the following steps:
1.
Order the I data points according to magnitude (either increasing or decreasing) so
· · SX1.
thatXf:;X 2 ~·
2.
Assign a probability to each datum. A convenient formula is P(Xi) = (i-1/2)//.
This assignment is consistent with the idea that the data are equally probable.
3.
Calculate the Zi value for each probability P(Xi ). This may be done using
Eq. (4-6e) or using a rational approximation to the probability integral given by
Z. = _
l
t
2.30753 + 0.27061 t
h r = ~ _2ln[P(X. )]
1.0 + 0.99229 t + 0.04481 t2 w e e t
z
for 0 < P (Xi )
~
0.5. This and a more accurate approximation are listed in
Abramowitz and Stegun (1965, p. 933). For P(Xi) > 0.5, use t =~ -2ln[P(Xi )]
and -Zi in the above formula (the Z's will be symmetrical about the point Z = 0).
4.
Plot the Xi versus the Zi. The points will fall approximately on a straight line if
they come from a normally distributed population.
This procedure should sound faintly familiar (Chap. 3); steps 1 and 2 are the same as
for computing the empirical CDF. That is because we are actually comparing the
empirical CDF with the normal CDF when we use a probability plot.
How far points on a probability plot may deviate from a straight line and still be
considered coming from a normal population is dealt with by Hald (1952, pp. 138-140).
Clearly, however, if there are few (e.g., fewer than 15) points, the variation can be quite
large while the underlying population PDF can still be considered normal. On the other
86
Chap. 4 Features of Single-Variable Distributions
hand, 100 or200 points should fall quite closely on a straight line for the population to be
considered normally distributed.
The
Log~Normal
Distribution
If errors multiply rather than add, the logarithms of the errors are additive. Since the
logarithm of a random variable is a random variable, the CLT may apply to the sum of
the logarithms to give a normal distribution. When this happens, the PDF of the sum is
said to have a log-normal distribution.
ExampleS- An Implausible Reason for Permeability To Be Log-Normally
Distributed. One of the most consistent assumptions regarding the
distribution of permeability in a naturally occurring permeable medium is that
it is log-normally distributed. One possible explanation for this is the theory
of breakage.
Suppose we start with a grain of diameter DpO• which fragments to a smaller
grain in proportionfo to yield a grain of diameter Dpl· Repeating this
process obviously leads to an immense tree of possibilities for the ultimate
grain diameter. However, if we follow one branch of this tree, its grain
diameter D pi will be
I
Dp1= ITfiDpi
i=O
which yields an additive process upon taking a logarithm. Thus, we should
expect ln(Dp1) to be normally distributed from the CLT. Since permeability
is proportional to grain size squared, it is also log-normally distributed.
The log-normal PDF can be derived from the basic definition of the PDF and the
standard log-normal variate:
ln X- ,ll]n X
z = -----''---'!.!-"'
O"lnx
Substituting this into Eq. (4-6c) gives the following for the CDF:
87
Statistics for Petroleum Engineers and Geoscientists
1{2ii XJexp [ - -1-(lntF(x;O"Jnx .UJnx) =
.UJnx) 2
2
'
O"ln x 2n: 0
20"lnx
l
dt
(4-7a)
and using y = ln X in the transformation given in Chap. 3, we have for the PDF
f(x; O"ln x, .U!n :J =
1- r:::.- exp [ - ~ln
x.U!n x)2]
2
0'
XO"Jn x\1 2n:
ln x
(4-7b)
The mean and variance of the log-normal distribution are given by
E(X) =
exp~
2
+ 0.5a x)
and
2
Var(X) = exp(2.Ux + 2a x)
The coefficient of variation is independent of f.lln
x
The PDF for the log-normal distribution is shown in Fig. 4-6. The significant features
of the PDF are the skewness to the left and the long tail to the right. Thus, we see that
most of the values in a log-normal PDF are small, but that there are a small number of
large values. Of course, the random variable X in the log-normal distribution cannot be
less than zero. Log-normal distributions appear to be at least as common in nature as
normal distributions.
The case of the logarithmic transformation is a common and interesting example of
how averages and nonlinear transforms can combine to produce nonintuitive results. If
the random variable, X, is log-normally distributed, then ln X- N(Jlx, O" ;). Therefore,
E(ln x) = J.lx. We might be tempted to assume that E(X) = exp[E(ln x)] = expC.ux). But, as
we saw above, the variability of ln x
also contributes to the mean:
2
E(X) = exp(Jlx)exp( a In x /2). This second term in the product can be significantly larger
than 1 for highly variable properties such as permeability, where a
commonly observed (see Example 2 of Chap. 10.)
a~x of around 4 is
Chap. 4 Features of Single-Variable Distributions
88
The logarithmic transformation case is just one example of how we must be careful
about using random variables in algebraic expressions. We will see later, especially in
the chapters on estimation and regression, that the deterministic manipulation of variables
that most of us learned in school is a special case. It is only in the special case that the
effects of variability can be ignored.
i
--><
-><
1.1..
0
2
0
Figure 4-6.
4
10
6
12
PDF and CDF for a log-normal distribution.
p-Normal Distribution
We can go one step farther and define a more general power transformation of the form
(Box and Cox, 1964)
Y={[(X+cP)-1]/p
ln(X +c)
p=O
which will render a wide range of sample spaces normal with the appropriate selection of
p and c. It is obvious that p = 1 is the normal distribution and p = 0 is log-normal.
89
Statistics for Petroleum Engineers and Geoscientists
Commonly, c is taken to be zero. The skewness of the distributions, intermediate
between normal and log-normal, corresponds to a value of p between 0 and 1. Figure 4-7
illustrates the shapes of the p-normal PDF.
Table 4-1 shows a few p-values estimated from permeability data from various media
types. There are also various types or measurements, and each data set has varying
number of data in it.
0.5
0.4
f(x)
0.3
0.2
0.1
0.0
0
10
5
15
20
X
Figure 4-7.
PDF shapes for various p.
Table 4-1. Estimates of p from various data sources. From Jensen (1986).
Data Set Label
Law
Sims Sand
Admire Sand
Lower San Andres
Pennsy1van ian
Nugget Sand
dst
plug
= drill stem test
core plug
Data Tyue
plug
plug
plug
dst
dst
plug
Number
of Data
48
167
330
112
145
163
1\
Media TyQe JLestimate, 12
sandstone
1.0
sandstone
0.5
sandstone
0.5
dolomite
0.1
limestone
0.0
sandstone
-0.3
Chap. 4 Features of Single-Variable Distributions
90
We see from this that p can be quite variable, especially in sandstone sediments. For
the carbonates in this table, the estimated p is always near zero. This reflects the greater
variability and/or lower average permeability in these sediments. Note also that p < 0 is
also possible, even though this does not fall within the range envisioned by the original
transform.
Besides the binomial, normal, and log-normal distributions, other PDFs are sometimes
used in reservoir characterization. For example, the exponential, Pareto, and gamma
distributions have all been used to estimate the number of small fields in mature
provinces (Davis and Chang, 1989). We will also meet the t-distribution in a later
chapter. It is clear that there are a large number of possible PDF's.
Uniform Distribution
This distribution is the simplest of all the"'commonly used CDF's and PDF's. It consists of
three straight lines (Fig. 4-8).
f(x) or F(x)
1.0
1
(a-b)
0
,---------~------~ f(x)
a
b
X
Figure 4-8.
The equation for the CDF is
PDF and CDF for a uniform distribution.
Statistics for Petroleum Engineers and Geoscientists
F(x; a, b) =
91
X<a
0
{ x-a
b _a
1
a 75. X 75. b
={b ~a
X<a
a-:5.X-:5.b
(4-8a)
b<X
and, consequently, for the PDF is
f(x; a, b)
(4-8b)
a<X
The uniform distribution is a two-parameter (a and b) distribution.The mean of this
distribution is E(X) =(a+ b)/2 and the variance is Var(X) = (b- aP/12. The popularity of
the uniform distribution arises from its simplicity and also from its finite bounds, X = a
and X = b. It is to be used when there is no a priori knowledge of the distribution type
but there are physical reasons to believe that the variable cannot take values outside a
certain range.
Triangular Distribution
The triangular distribution is the next step up in complexity from the uniform distribution.
Its PDF and CDF are shown in Fig. 4-9.
The triangular distribution is a three-parameter distribution (a, b, and c). The equation
for the PDF is
X<a
0
f(x; a, b, c) = (
__1_)
c-a
x-a
b-a
c -X
c-b
a~X~b
(4-9)
b<X <c
c<X
0
with mean E (X) = (a + b + c)/3. The variance of this distribution is (McCray, 197 5)
Var(X) =
(c- a)2- (b- a)(c- b)
18
Chap. 4 Features of Single-Variable Distributions
.92
As for the uniform distribution, the triangular distribution is useful when there is :feason
to confine the variable to a finite range. However, it frequently occurs that there is a most
likely value within this range, but still no a priori knowledge of the distribution form. In
these cases, the triangular distribution is appropriate. As Fig. 4-9 suggests, the triangular
distribution can be used as an approximation to both the normal and log-normal
distributions.
1
. F(x)
-E...
0
~
LL
f(x)
0
a
0
b
c
X
Figure 4-9.
PDF and CDF for a triangular distribution.
Exponential (Boltzmann) Distribution
The exponential distribution is shown in Fig. 4-10. This distribution is a one-parameter
function with the following form for the CDF:
F(x; A..)= 1 - exp(-x/A..),
x>O, A..> 0
(4-IOa)
and for the PDF
1
f(x, A..)=~ exp(-x/A..),
(4-IOb)
The mean and variance are A.. and A,2, respectively. Despite its simplicity, the exponential
distribution is rarely used in reservoir characterization. However, its slightly more
Statistics for Petroleum Engineers and Geoscientists
93
complicated cousin, the Gibbs or Boltzmann distribution, is used extensively in
combinatorial optimization schemes (Chap. 12). The feature desired in these applications
is that the mode of the exponential distributions occurs at the smallest value of the
random variable.
1
-><
-><
....
0
u..
0
0
Figure 4-10.
X
PDF and CDF for an exponential distribution.
4-5 TRUNGA'IE::JtDAIASETS
A truncated sample occurs when data values beyond some limit are unsampled or
unmeasurable. Truncation can be the result of restrictive sampling, deficient sampling, or
limitations in the measuring device. For example, some apparatus for measuring
permeability cannot measure values less than 0.1 mD. Adata set that is missing largevalued samples is top-truncated; if it is missing small-valued samples it is bottomtruncated. The upper plot in Fig. 4-11 illustrates these concepts with the random variable
mapping outside of a limiting value, X lim .
Before proceeding, a word is in order about terminology. A data set is truncated if the
number of unmeasured values is unknown; it is censored if they are known. Truncation
in this section applies only to data; when applied to theoretical (population) CDF's,
truncation can mean that a portion of the variable range is not attainable on physical
grounds (porosities less than zero, for example).
Chap. 4 Features of Single-Variable Distributions
94
As always in statistical manipulations, we cannot infer the underlying reason for the
truncation (at least not from the statistics alone); we can only detect its occurrence and
then attempt to correct for it. The correction itself is based on an application of the
definition of conditional probabilities given in Chap. 3. See Sinclair (1976) for more
details.
Truncation
Xiim
Mixed
Populations
00
+
Figure 4-11.
Schematic plots of truncation (upper) and mixed populations (lower).
The basic tool is the CDF, F(x), or the complementary CDF, Fc(x). We assume that
the untruncated sample CDF is normal or can be transformed to be normal; the data sets
are log-normal here. Recall that the horizontal axes in both plots are probabilities;
95
Statistics for Petroleum Engineers and Geoscientists
Prob(X :s; x) in the CDF and Prob(X;:::: x) in the complementary CDF. We use F and Fe in
percent in this section.
Figure 4-12 shows the effects of bottom and top truncation on a sample CDF.
Omitting the data points, we show only lines in this figure.
----;-------·;--------r-··t····t··-;-----i·--·-·J··;--·-·J··-;------- · ------1····
l 25fTf"KF~,. ·:·t
····:·······-~---··---r·-·1····-~---;...
: ! j :
ooOO:••••••••t•••••t•••i••
····1········1.
0,01
.01
.1
-~
ri···
··· · -:·····. ---~---·····f···-····1····
0
:
:
u,nt~unc,ated:
•j~•·•:•••ooti•••oot···i•ooooooo(••••"t"
···t··+ 1sf t~p ~:ru:nc~t~d t······-j--··
5 10 20 30 &I 7080
so 95
99
99.99
99.9
Cumulative Percent Smaller than Value
Figure 4-12.
Schematic of the effects of truncation on a log-normal CD F.
Truncation evidently flattens out the CDF in the region of the truncation. In fact, it is
relatively easy to see the truncated value (xum in Fig. 4-11) by simply estimating on the
value axis where the leveling out takes place. What is not so easy to see is what
percentage of the data values are truncated, but two observations are pertinent.
1.
The slope of the CDF (related to the variance) at the extreme away from the
truncated region is essentially the same as for the untruncated sample.
2.
The probability coordinate Fe where the leveling occurs is roughly the percentage
of the data points that are omitted by the truncation.
In order to correct for truncation, we must somehow determine the untruncated CDF.
For the CDF, this process is accomplished by the following manipulations:
Top truncation
F =Fa( I - ft)
(4-lla)
Bottom truncation
F = fb + Fa(l - fb)
(4-llb)
96
Chap. 4 Features of Single-Variable Distributions
In these equations, F is the untruncated CDF, Fa the truncated (apparent) or original
CDF, andfb and.ft are the fraCtion of data points bottom- and top-truncated, respectively.
From Eq. (4-12a), we have F <Fa and from Eq. (4-12b) F >Fa As mentioned above,Jb
(or ft) can be approximated by simple inspection of where the CDF flattens out.
However, to be more certain severalfb (or ft) values should be chosen and F vs. X plotted;
the correct degree of truncation fb (or ft) is that which gives the best straight line. The
similar relations for the complementary CDF apply and are left to the reader as an
exercise.
Despite the trial-and-error nature of the process, determining the amount of truncation
is fairly quick, especially on spreadsheet programs. Figure 4-13 illustrates a bottomtruncated data set of original oil in place from a region in the Western U.S.
Truncation
. . .
... ... ... .. ...
.. . ..
'.
.' ..'' ..'' ..' ..'' ..'' ...'
==
-
t"\1
0
0
Ill
-!::
0
'
'
'
'
... : --····· ..:.... -.. ! .. ,; ____ : ...:.. .. -. .!_ •••• ~---~---.:. •• -~ ------~---- ••• -~ •••
l
~
l
i
l
~
1 1 1 1
1
1 1
-r~A+O)
t
l
1
!!!!iii~!!!
c: ·-
·o,=
·- .D.
0
.....
l l 1 l 1 l ~qe(
1· ;::( ....
P~l
r
Percent Smaller than Value
Figure 4-13.
CDF for original oil in place showingfb in percent.
For this case, the reason for the truncation is apparent: reservoirs smaller than about
0.5 billion barrels were simply uneconomical at the time of survey. However, the value
offb from Eq. (4-12b) required to straighten out the data indicates that approximately
40% of the reservoirs are of this size or smaller. The corrections leave the untruncated
portion of the curves unaffected. The following example indicates how a truncation
correction can affect measures of central tendency.
Example 6- Top Truncation of Core Permeability Data. A waterflood in a
particular reservoir invariably exhibits greater injectivity than is indicated by
97
Statistics for Petroleum Engineers and Geoscientists
the average core penneability data. Among the many possible causes for this
disagreement is top truncation in the core data.
The first column of Table 4-2 below shows an abridged (for ease of
illustration in this exercise) set of permeability data from this reservoir. The
second column shows the Fa= lOO[(i- t)!I] values, where i = 1, ... , I is an
index of the data values and I is the total number of data points. The set is
clearly top truncated, as shown in Fig. 4-14.
Table 4-2. Data for Example 6. Columns 3 and 4 show F values that have
been corrected forft = 0.2 and 0.6 using Eq. (4-12a).
Permeability
mD
0.056
0.268
0.637
1.217
2.083
3.343
5.149
7.722
J 1.389
16.641
24.243
35.423
52.244
78.358
Fa
Original
F
F
ft = 0.2
ft = 0.6
3.570
10.714
17.857
25.000
32.143
39.286
46.429
53.571
60.714
67.857
75.000
82.143
89.286
96.429
2.857
8.571
14.286
20.000
25.714
31.429
37.143
42.857
48.571
54.286
60.000
65.714
71.429
77.143
1.429
4.286
7.143
10.000
12.857
15.714
18.571
21.429
24.286
27.143
30.000
32.857
35.714
38.571
Theft= 0.6 curve (60%) has been overcorrected, as evidenced by the upward
curvature. The best correction is slightly less thanft = 0.2 (20% ). But, taking
ft = 0.2 to be best, the geometric mean of this data set (the median) has been
increased from around 6.5 mD to about 12 mD. This correction may be
sufficient to explain the greater-than-expected injectivities. We should seek
other causes to explain why more than 20% of the data are missing from the
set.
A data set may be both top- and bottom-truncated. In this case we have
Chap. 4 Features of Single-Variable Distributions
98
(4-12)
Using Eq. (4-13) requires a two-step trial and error to determine bothft and fb· The
reader may verify the analogous expression for P:.
100
10
Q
E
>:
~
..0
«<
(I)
E
,_
(I)
ltl..
0.1
0.01
.01
.1
5 10
20 30
50
70 80
90 95
99
99.9 99.99
Percent Smaller than Value
Figure 4-14.
4~6
CDF illustrating correction for top truncation.
PARTITIONING DATA SETS
Nature is not always so kind as to give us data sets from only single PDF's. See the lower
plot in Fig. 4-11 for a schematic illustration. In fact, many times each reservoir contains
mixtures of materials with different parent distributions. The aggregate distribution is
what we see when the reservoir is sampled. When this happens, the power of the PDF
and the CDF gets distinctly blurred, but we can still gain some insight from them.
Partitioning is the separating of a mixed distribution function into its component
(parent) parts. Partitioning is most efficient when the data values separate naturally into
groups on the basis of variables other than the variable in the CDF. This is one of the
primary roles geology serves in reservoir characterization.
When we partition, we obtain more homogeneous elements that we expect to be able
to understand better. For example, in Example 4 we showed how combining two
Statistics for Petroleum Engineers and Geoscientists
99
elements increases variability. If we know the variability of each element and its
probability of occurrence, we can develop a moqel of the total variation that is more
realistic than one that is based only on the mean and variance of the aggregate. ·
·In the absence of geological information, partitioning is usually difficult, especially
when the CDF consists of more than two parent populations. For these reasons, we limit
the treatment here to partitioning into only two CDF's. Many of the basic assumptions
here are the same as in the discussion on truncation: the parent populations are lognormally distributed and all operations will be on the cumulative distribution function
(CDF) or its complement. We further assume that the parent populations are themselves
·
untruncated.
Let the sample CDF of parent distributions A and B be FA (x) and Fn(x), respectively.
Then the probability coordinate of the mixed CDF is given by
(4-13)
where fA andfB are the fractions of the samples (data values) taken from populations A
and B. This intuitive rule is actually a consequence of both the additive and
multiplication rules for probabilities. It implies that data are selected for incorporation
into F without regard to their value (independence) and a sample cannot be in both
distribution A and distribution B (mutually exclusive). Of course, we must have
fA +in= I.
Equation (4-13) is also a rule for combining the F's on the horizontal axis of the CDF.
For example, ifFA= 0.9 and FB = 0.3 for a particular value of x (recall X< x) and the
mixture is 30% A and 70% B, then F = 0.48 of the samples in the mixed CDF are less
than x. Of course, this mixing is not linear on the probability axis of the CDF. The
essence of partitioning is actually the reverse of this: determine fA andfB given F(x) and
some means to infer FA and Fn. Following Sinclair (1976), there are two basic ways to
partition based on whether populations A and Bare nonintersecting or intersecting.
Nonintersecting Parents
Nonintersecting means that the parent populations overlap very little. Figure 4-15 shows
three mixed CDF's for parents A and B (A has the smaller variance) as a function of
various proportions of A.
The mixed CDF has the following attributes:
100
Chap. 4 Features of Single-Variable Distributions
1.
The central portion of the curve is the steepest. This observation is what
distinguishes nonintersecting and intersecting CDF's. The steepness increases as
the overlap in X values between A and B decreases.
2.
The F value of the inflection points roughly corresponds to fs = 1 -fA.
3.
The extremes (F approaching zero or one) become parallel to the CDF's for the
parent samples.
CD
:J
11:1
>
Percent Smaller than Value
Figu~
4-15.
CDF's for nonintersecting parent populations.
These observations .suggest the following graphical procedure for partitioning a
sample drawn from two nonintersecting populations:
1.
Plot the data points F vs. X and draw a smooth line through the points on
probability coordinates. The smooth line represents the mixed population CDF.
2.
Draw in straight lines corresponding to the parent populations as suggested in
Fig. 4-15. The parent population with the smallest variance (slope) will be nearly
tangent to the mixed CDF at .one of the extremes. This will also be true of the
parent population with the largest variance, but less so.
3.
Estimate the partitioning fraction from the inflection point on the smooth line.
4.
The fA andfs are now known from step 3; FAandF8 from step 2. Use Eq. (4-14)
to reconstruct F.
Statistics for Petroleum Engineers and Geoscientists
101
If this F(X) agrees with the original data, all inferred values are correct; if not, adjust
the partitioning fraction and return to step 3. If it appears that a satisfactory match cannot
be attained, it may be necessary to adjust FA and/or F B In general, convergence is
attained in three to five trials as the following example shows.
Example 7- Partitioning into Two Populations. The data set of 30
permeability values in Table 4-3 is believed to come from two parent
populations.
The first column contains the permeability values, and the third and fourth
columns the FA and FB values. These are the X coordinates of the two
population lines in Fig. 4-16 corresponding to the permeability values in
column one. Columns five and six contain F for two estimates offA.
The shape of the original experimental CDF in Fig. 4-16 clearly indicates a
mixed population.
Even though the X coordinate of the inflection point suggests fA= 0.7, we use
fA =0.5 and 0.8 for further illustration. However, neither fA value yields a
particularly good fit to the data (the solid line in Fig. 4-16), probably because
the line for FB is too low (mean is too small). For accurate work, FE should
be redrawn and the procedure repeated. However, the procedure is sufficiently illustrated so that we may proceed to the second type of mixed
populations.
Q
E
>:
=
..0
ill
10
G)
E
...
G)
D.
1
~~~~~LL~LL~~~~~
.001 .1
.01
s
1
2>
10
ro eo
3)
95
99.9 99.999
70 90 99 99.99
Percent Smaller than Permeability
Figure 4-16.
Experimental CDF for Example 7.
102
Chap. 4 Features of Single-Variable Distributions
Table 4-3. Data for Example 7.
Permeability
mD
1.70
1.75
1.80
1.90
1.95
1.95
1.95
2.00
2.10
2.10
2.10
2.15
2.15
2.15
2.20
2.25
2.30
2.50
2.60
2.70
2.70
3.50
4.95
8.10
11.50
14.50
15.50
16.50
22.50
28.00
F
original
1.67
5.00
8.33
11.67
15.00
18.33
21.67
25.00
28.33
31.67
35.00
38.33
41.67
45.00
48.33
51.67
55.00
58.33
61.67
65.00
68.33
71.67
75.00
78.33
81.67
85.00
88.33
91.67
95.00
98.33
FA
FB
estimated estimated
3.00
6.00
9.00
12.00
15.00
45.00
46.00
51.00
54.00
57.00
60.00
63.00
66.00
69.00
72.00
78.00
85.00
91.00
94.00
96.00
99.00
99.99
99.99
99.99
99.99
99.99
99.99
99.99
99.99
99.99
0.80
0.90
1.00
1.00
1.00
1.00
1.00
1.00
1.50
1.50
1.50
2.00
2.00
2.00
3.00
3.40
3.50
3.90
4.00
4.00
4.50
8.00
20.00
50.00
70.00
80.00
85.00
86.00
95.00
98.00
F
fA= 0.5
F
fA=0.8
1.90
3.45
5.00
6.50
8.00
23.00
23.50
26.00
27.75
29.25
30.75
32.50
34.00
35.50
37.50
40.70
44.25
47.45
49.00
50.00
51.75
54.00
60.00
74.99
84.99
89.99
92.49
92.99
97.49
98.99
2.56
4.98
7.40
9.80
12.20
36.20
37.00
41.00
43.50
45.90
48.30
50.80
53.20
55.60
58.20
63.08
68.70
73.58
76.00
77.60
80.10
81.59
83.99
89.99
93.99
95.99
96.99
97.19
98.99
99.59
Statistics for Petroleum Engineers and Geoscientists
103
Intersecting Parents
Here the parent populations overlap to a significant degree. The signature for intersecting
parents is not as clear as for nonintersecting parents; however, the differences in their
respective CDF's is dramatic. Compare Figs. 4-15 and 4-17.
Just as for nonintersecting parents, partitioning these CDF's requires a trial-and-error
procedure using Eq. (4-13). However, there is no longer an inflection point to guide the
initial selection offA· Instead, the central portion of the curve is flatter than the extremes,
a factor that makes partitioning of this type of mixture more involved. Nevertheless,
following Sinclair (1976), there are some general observations possible from Fig. 4-17.
1.
The central portion of the mixed CDF is flatter than the extremes; all three mixed
curves intersect each other within this region. This is what makes the inflection
point difficult to identify.
2.
The value range (on the vertical axis) of the central segment is greater than the
range of the parent population having the smallest range (sample A in Fig. 4-17).
3.
The F range (on the horizontal axis) of the flat central segment is a coarse
estimate of the proportion of the small-range population in the mixture lfA in
Fig. 4-17).
1000
100
10
.01
.1
5
10
20 30
50
70
so
90 95
99
Percent Smaller Than Value
Figure 4-17.
CDF's for intersecting populations.
99.9 99.99
104
Chap. 4 Features of Single-Variable Distributions
Observation 2 helps in drawing the CDF's for one of the parent populations;
observation 3 gives an estimate of fA- However, estimating the properties of the second
parent CDF adds another level to the trial-and-error procedure.
Although the corrections outlined above are based on probability laws, the lack of a
physical base limits the insights to be derived from these procedures. In particular, it may
not be clear if corrections are needed-deviations from a straight line might be caused by
sampling error. Furthermore, deciding when a given degree of correction is enough is a
matter of judgment. Sinclair (1976) recommends that, once a given CDF is partitioned,
the practitioner should look elsewhere (more or other types of data, usually) for an
explanation. For example, suppose that two sources of sediment feed a river that later
deposits sand downstream. If the two distributions of, say grain size, are each normally
distributed with the same variance but different means, then the deposited sand may have
a sample PDF that has two peaks. Generally speaking, we can expect that mixtures of
material will originate from parents having distributions with neither the same means nor
variances.
Another uncertainty is the underlying assumption that the untruncated or parent
populations are themselves transformed Gaussian distributions. As we have seen, there is
little physical base for a given distribution type to prevail. As in most statistical issues,
these types of decisions depend to a great extent on the end use.
Finally, neither the CDF nor the PDF say anything about the relationship of one
sample space to another or about the spatial arrangement of the parameters. This is
because both treat the observations as being ordered (sorted) without regard to where they
came from. We will have something to say about such relations in Chaps. 8 to 11 that
deal with correlation and autocorrelation.
Given these limitations, simply plotting the sample CDF's often gives diagnostic
information without further analysis. For example, Fig. 4-18 gives schematic plots that
represent various "type" curves for some common complications. Many data sets can be
quickly classified according to this figure and such classification may prove to be
sufficient for the problem under consideration.
Statistics for Petroleum Engineers and Geoscientists
Bottom truncated
105
Top truncated
F
F
Top and bottom truncated
Non-normal x
X
F
F
Two parentsNon-intersection
Two parentsIntersecting
X
F
Figure 4-18.
4-7
F
"Thumbprint" CDF's for truncated and mixed populations.
CLOSING COMMENT
Distribution functions provide the bridge between probability and statistics. The ideas of
distribution functions arid, in particular, the ways of summarizing them through the
expectation and variance operations will recur in many places, as will the Central Limit
Theorem. The most important ideas deal with the manipulations of the expectation, as
these are central to the notions of covariance and correlation. However, the reader should
1
106
Chap. 4 Features of Single-Variable Distributions
be reminded from the last few example PDF's that, even though statistics is a flexible and
powerful tool, its benefit is greatly enhanced with an understanding of underlying
mechanisms.
5
Estimators and Their
Assessment
Since defming probability in Chap. 2, we have begun to appreciate the difference between
the population and the sample. The population represents the entire body from which we
could draw examples and make measurements. The sample is a limited series of
measurements and represents the best we can do, given the physical and fiscal constraints
of acquiring data.
We sample to assess the properties of the population. The assessment procedure is
called estimation; there is a deliberately implied uncertainty in the name. The ultimate
would be to obtain the population PDF; then we could calculate whichever parameters we
sought. We have only a sample, however, and parameters must be estimated from this
set of data based on the results of a limited number of experiments (Chap. 2). We also
want to know, once we have an estimate in hand, how well it represents the true
population value. This is where confidence limits are helpful.
A simple example may help to highlight the issues in estimation. Suppose that we
decide to calculate the arithmetic average of five porosity values from similar rock
volumes: 21.1, 22.7, 26.4, 24.5, and 20.9. Since the formula for the arithmetic average
is
-
1 I
f/J=- 'L l/Ji
I i=l
107
Chap. 5 Estimators and Their Assessment
108
where I is the number of data, we obtain if= 23.1. We have a number. But, what does
the value 23.1 represent? What have we calculated and what does it represent? Is 23.1
any "better" than 21.1, 24.5, or any of the other numbers? At least 21.1, 22.7, etc. are
measured values, whereas there may not be any part of the reservoir with q, = 23.1. As
we shall soon see, this procedure has several assumptions that have gone unstated.
Consequently, the value 23.1 may or may not be suitable for our purposes.
5=1
THE ElEMENTS OF ESTIMATION
Estimation involves four important elements:
The population quantity to be estimated, 9, (e.g., arithmetic mean, standard
deviation, geometric mean) from a set of data. () can represent one or more
parameters. If it represents more than one parameter, we can consider it to be a
vector instead of a scalar.
2. The estimator, W. This is the method by which the data values will be combined
to produce the estimated or sample quantity.
3. The estimate, ~.probably the most familiar component to the reader.
4. The confidence interval or standard error, [Var(~]l/2. This allows us to judge
how precise is the estimate ~ .
1.
In the porosity example, we chose the arithmetic average for the one-parameter
estimator Wand 23.1 = ~. No confidence interval was calculated. Furthermore, the most
important step-choosing 9--was completely ignored.
The number of parameters required depends on the form of the PDF and the needs of
the user. Recall from Chap. 4 that the binomial distribution has a two-parameter PDF:
namely, p and n, the probability of success in each trial and the total number of trials,
respectively. In many experimental situations, it is advantageous to estimate p from a
limited number of experiments with n trials each and then use this estimate to characterize
all future experiments. In that sense, p, the estimate of the population value p, is an
average probability of success for a particular type of experiment against which other
similar experiments may be judged. On the other hand, we may have an application
where we only need the variance of the distribution, which is np(l-p), without explicitly
needing to know what p is. In this case, we may decide to estimate p-a parameter of the
PDF-or we may decide to estimate the quantity p(l-p) directly.
Similarly, the normal distribution has two parameters, the mean and variance, which
may be estimated to characterize a random variable suspected of being normally distributed
(e.g., porosity). If a physical process is normally distributed, the mean and variance
Statistics for Petroleum Engineers and Geoscientists
109
precisely describe the PDF. Other distributions may require more (or fewer) parameters or
statistics.
Recall that a statistic is any func.tion of the measured values of a random variable
(Chap. 2). It is an estimate of a parameter that could be computed exactly if the PDF
were known. To distinguish between the estimate and the estimated quantity, the
estimate is often prefixed with the terms empirical or sample. For example, some
common statistics are measures of central tendency (e.g., averages or sample means) and
measures of dispersion (e.g., sample variance or interquartile range), but they are by no
means limited to these. As we will see, there may be several ways to estimate a
parameter.
5-2
DESIRABLE PROPERTIES OF ESTIMATORS
Since any given statistic (parameter estimate) is a function of the data, then it too is a
random variable whose behavior is described by a PDF. That is, given an estimator W
and a set of data {XI.X2, ... ,XJ} taken from a population with parameter value 8, W
produces an estimate ~based on the data: ~ = W(XI. X2, ...,XJ). Knowing this~ there are
several features that a good estimator will have: small bias, good efficiency, robustness,
and consistency. In addition, it should produce physically meaningful results.
5-3
ESTIMATOR BIAS
The PDF of an estimate ~ should be centered about the population parameter () we wish
to estimate (Fig. 5-1). If this is not true, the estimator W will produce estimates that
tend to over- or underestimate 9, and the estimator is said to be biased. Bias is given by
the expression
b = E(fJ}- ()
Bias is generally undesirable, but sometimes an esti>mator can be corrected for it. Bias has
two principal sources, measurement resolution and sampling, as the following example
illustrates.
Example 1 - Biased Sampling. Data from core-plug samples are sometimes
misleading because of preferential sampling. The samples may not be
representative for a variety of reasons: incomplete core recovery (nonexistent
samples), plug breakage in friable or poorly consolidated rock (failed
samples), and operator error (selective sampling).
1
i
Chap. 5 Estimators and Their Assessment
110
1\
E(e)
1~
Figure 5-1.
e
b
•I
Estimator bias (b) depends upon the PDF of~.
An interval of core has two lithologies, siltstone and sandstone, in the
proportions 30% siltstone and 70% sandstone. C/Jsilt = 0.10 and
C/Jsand =0.20, If the siltstone is not sampled, what is the bias in the
average porosity obtained from plug samples?
The true mean porosity is given by
cfrr = 0.30. 0.10 + 0.70. 0.20 = 0.17
All samples are sandstone with one porosity value. Therefore, ignoring
measurement error, the apparent formation porosity is given by
C/Ja = 1.0 • 0.20
=0.20
i
Statistics for Petroleum Engineers and Geoscientists
111
The bias is b<P = l/Ja- ¢T = +0.03, or about 18% too high.
5-4
ESTIMATOR EFFICIENCY
The second property of a good estimator is that the variability of its estimates should be
as small as possible. An estimator with a small variance is said to be precise or efficient.
Do not mistake the variance of the sample space for the variance of the estimator. The
former is a property of the physical variable and the latter is a property of the data and the
estimation technique (Fig. 5-2).
/-arorw
Figure 5-2.
Estimator efficiency is based on how variable the estimates~ are.
[Var(~]l/2, the standard deviation of the estimates, is called the standard error. There
are theoretical statements such as the Cramer-Rao Inequality (Rice, 1988, p. 252) about
the minimum variance an estimator can attain. When an estimator has no bias and this
minimum variance, it is said to be an MVUE (minimum variance, unbiased estimator).
Example 2a- Determining Uniform PDF Endpoints (Method 1). Monte
Carlo reserves estimation often assumes a uniform PDF for one or more
Chap. 5 Estimators and Their Assessment
112
variables (Chap. 4). How do we obtain the lower and upper limits?
Obviously, we must depend on the data to reveal these endpoint values.
In this example, two methods for estimating endpoint values are presented and
compared for bias and efficiency. Rohatgi (1984, pp. 495-496) presents a
third method. To simplify the discussion, we assume that the upper endpoint
is unknown and the lower endpoint is zero. In the context of the previous
discussion on estimators, eis the upper endpoint and the parameter to be
estimated (Fig. 5-3).
For the uniform PDF, the mean value of the random variable X is given by
8
00
E(X) =
Jxf(x)
-oo
dx = j(x/0) dx = 012
0
That is, the arithmetic mean is exactly one-half the upper limit 0 when the
lower limit is zero (Fig. 5-3).
We use this result to provide an estimator for
e.
f(x)
1
9
0
9
2
Figure 5-3.
e
Uniform PDF with lower limit 0, upper limit 0, and mean 0/2.
Given I samples of X ,X1. X2•... ,XJ, taken from a uniform distribution, the
above suggests that twice the arithmetic average should give an estimator for ():
Statistics for Petroleum Engineers and Geoscientists
1 i3
2 I
~1 = 2 X=/~ Xi
1=1
Let W 1 be this estimator (the subscript 1 on e, W, and b below refers to the
first estimator). The bias of this estimator is as follows:
Since b1 is zero, the estimator W1 is unbiased. The variance of W1 is
Var(~l) = E[(~l- (J) 2] =E(~l2) _ (J 2
= 4[E( X)2- E( X)2] = 4Var( X)
-
1
Since (see Eq. (4-4)) Var( X)= I Var(X) for I independent samples Xi,
[ix
=7
2IO )dx - (
o2t4)] =o2!3I
The variability of the estimates ~1 depends upon(} and/, the number of data.
Var(~ oc J-1 is a very common dependence, i.e., the standard error is halved with a
four-fold increase in data-set size. This suggests that overly stringent specifications for
estimates can be quite costly. Var(~ oc () 2 is reasonable since, with lower limit fixed at
0, the upper limit (}will control the variability of X.
Example 2b- Determining Uniform PDF Endpoints (Method 2). A more
obvious estimator than W 1 for the upper limit of a PDF is to take the
114
Chap. 5 Estimators and Their Assessment
maximum value in the data set: ~ = Xmax· Let W2 be defined as
~2 = max{Xl>Xz, ... ,XJ}.
From theory based on order statistics (Cox and Hinkley, 1974, p. 467),
1
E(~)= (/+1
-)e
Hence, W2 is a biased estimator with b2 = -l/(1+1). As we might expect,
Xmax never quite reaches e, no matter how many data we have. We can
correct for this bias by redefining W2 as [(1+1)/1] max{X 1. X2, ...,XJ}. This
estimator is unbiased because E(~) = 8. It has variance
If we compare the variances of W 1 and W2· we conclude that W 2 is more
efficient than W1 because W2 has a a lower variance than W1. Their relative
efficiencies are
A similar procedure could be used to find an estimator for the lower limit a of
a uniform PDF: ~= [!1(1+1)] min{X1,X2, ... ,XJ}.
The estimator, W 2. in Example 2b was biased but more efficient. Efficiency is
evidently not an intuitively obvious property. W 1 explicitly used all the data in the
estimator, whereas W2 only used the values to order the data and explicitly used only the
maximum. It may appear that W2 made less use of the information contained in the data,
but that is not the case.
There are several methods to develop estimators (Rice, 1988, Chap. 8). The methods
of moments and of maximum likelihood are two examples. Maximum likelihood
requires that the population PDF be known (or assumed). This additional information
helps to make likelihood estimators efficient.
Combining the properties of estimator bias and variability introduces an important
property known as consistency. An estimator W is consistent if, as the number of data,
l, becomes large, ~ approaches e (asymptotically unbiased) and the variance of ~
approaches zero.
Statistics for Petroleum Engineers and Geoscientists
5-5
ESTIMATOR
115
ROBUSTNESS
Estimator robustness is the ability of the estimator to be relatively uninfluenced by errors
in a small proportion of the data set. We appreciate that errors can occur in the
measurement and recording of data; a decimal point could get moved or a specimen could
be improperly tested. If the unrepresentative values have a large influence on the estimate
produced, then the estimator is not robust. We will always have problems trying to find
an estimator that is responsive to meaningful data values but is insensitive to wild,
unrepresentative data.
Consider, for example, the following set of porosity data: 8.3, 9.6, 7.4, 8.9, and 9.1.
For this set, X= 8.7 and~o.5o = 8.9, which are in good agreement (within 2%). What
if one of the data values, 9.1, were misrecorded as 19.1? In that case, X= 10.7
happens
1\
while Xo.so = 8.9, a disagreement of 18%, which can be significant.
Substantial work has been done concerning robust estimators. Barnett and Lewis
(1984, Chap. 3) discuss some quantitative assessments for robustne~. A simple example
is using the median to estimate the mean. The sample median (X 0.50) is the middle
value in a set of I data:
X(l+l)/2
1\
Xo.5o =
{
1
2 [XI/2 + X (1+2)/2]
for I odd
for I even
where X 1:::;; X2::;; X3::;; ···::;;XI is an ordered data set. If the data come from a normally
distributed population with mean J.l, ~0.50 is an unbiased estimate of Jl, but it is about
25% less efficient than the arithmetic average, X (Kendall and Stuart, 1977, pp. 349350). Frequently, the most efficient estimator is less robust than other estimators. The
more efficient estimators use all information, including knowledge of the population
PDF, but they can deteriorate quickly with small deviations from the underlying
assumptions.
We will not dwell further on robust estimators because there are excellent treatments
elsewhere (e.g., Hoaglin et al., 1983; Barnett and Lewis, 1984). We will, however,
sometimes refer to an estimator as robust or nonrobust. In reservoir description, we must
examine the robustness issue from two perspectives: choosing a robust estimator when
we have a choice and, when we do not have a choice, assessing the estimator sensitivity.
116
5-6
Chap. 5 Estimators and Their Assessment
ASSESSING THE SENSITIVITIES OF
ESTIMATORS TO ERRONEOUS DATA
Quantification of estimator robustness is often difficult. We can, however, make some
progress by looking at the differential sensitivities for some estimators. If we have an
analytic expression for the estimator W, then we can calculate the sensitivities based on
the [email protected]= W(X1, X2, .. .,XJ):
where [email protected] is the error in the estimate arising from the error AXi in the measurement of
the ith datum. This expression comes from the truncated Taylor series expansion of W
about the true data values XI,X2, ... ,XJ, so it may only be useful for small errors (less
than 20% or so) in the X's. The expression also assumes the AX/s are independent.
Given that this approach has its limitations, it is often still useful to examine the
estimator sensitivities to measurement errors. Knowing these sensitivities, resources of
data acquisition can be better applied to give the least erroneous estimates possible.
Example 3 -Archie's Law Sensitivity Study. We want to assess hydrocarbon
saturation in a formation by measuring formation porosity and resistivity
with wireline logs and cementation and saturation exponents in the
laboratory. We use a model, Archie's law,
to predict water saturation, where a is a constant, m and n are the cementation
and saturation exponents, respectively, and Rw and Rt are the water and
formation resistivities (Archie, 1943). In this case, our estimator-Archie's
law-is fixed, but we want to know whether to pay for the expensive logging
suite (accurate to ±5%) or spend extra money acquiring better laboratory data.
We assess the impact of measurement errors in a, ¢!, m, n, Rw, and Rt upon
the water saturation Sw. Taking partial derivatives of sums and differences is
easier than differentiating products and quotients. Since Archie's law consists
only of products, quotients, and exponents, we can simplify things by taking
logarithms of both sides before calculating the partial derivatives:
1
ln(Sw) =- [ln(a) + ln(Rw)- mln(¢1)- ln(Rt)]
n
Statistics for Petroleum Engineers and Geoscientists
117
Hence, in the case of the saturation exponent n, for example, we have,
assuming that a, cp, m, n, Rw, andR1 are independent,
aln(S
)
1
dn. w =- n-2 [ln(a) + ln(Rw)- mln(</J)- ln(R =-;,
1)]
ln(Sw)
asw = _ Sw ln(S )
an
n
w
or
Hence,
..1Sw
Sw
=- -ln(Sw)..1n
n
A similar approach for the other variables gives the following expression for
the total error in Sw arising from errors in all the measured variables:
..1Sw
Sw [
=--;;- ln(Sw)..1rt + ..1Rw
Rw -
..1R t ..1a
Q_!P.
Rt + --;;- m <P - ln(</J)..1m
J
The proportional change in water saturation is
..1Sw
Sw
=-;;,1 [.-ln(Sw)..1n + L1Rw
Rw -
..1R t ..1a
Q.!P.
Rt. + --;;- m <P - ln(</J)..1m
J
Thus, Rw. R 1, a, and <P all contribute proportionately to the total error ..1Sw.
If m =2, porosity errors are twice as significant as resistivity errors. On the
other hand, m and n contribute directly according to the porosity and water
saturation of the material under consideration. Errors in n could lead to
significant errors in Sw at low water saturations, while errors in m can be
important for low-porosity media. Thus, extra money expended to make
careful laboratory measurements may be worthwhile.
Unlike Archie's law, which deals with static quantities, there are also instances where
measurements can influence the predicted flow properties of the reservoir. Relative
permeability measurements are a good example of this situation. They are required for
immiscible multiphase flow. They are a laboratory-determined property and are usually
based on results from a few, small rock samples. Thus, these data are susceptible to
errors and those errors may be important, as described in the next example.
Chap. 5 Estimators and Their Assessment
118
Example 4 -Assessing the Effects of Relative Permeability Errors. In a
common case of one spatial dimension and two phases (oil and water), the
frontal advance solution (Willhite, 1986, pp. 59-64) of Buckley and Leverett
shows that the fractional flow is the important quantity that relates the
relative permeabilities to flooding performance. It is not the relative
permeability measurement errors per se that matter, but the effects of these
errors upon the fractional flow. If we ignore gravitational effects and assume
negligible capillary pressure, the water fractional flow is
1
fw = -----='----
1+
kro
Jlw
krw f.lo
where kr is the relative permeability, 11 is the fluid viscosity, and the
subscripts o and w refer to oil and water. By definition fw :s:; 1 for this case.
An analysis similar to that of Example 3 shows that
Nw = (1- fw) (Llkrw _ Llkro)
fw
krw
kro
Thus, errors infw are less than errors in the relative permeabilities, since
0 < (1- fw)< 1. If a shock front develops, only fw values exceeding the
breakthrough valueJwbt. exist (see Lake, 1989, Chap. 5). This further
mitigates the impact of relative permeability errors. The errors in kro. which
can be large when kro is small, produce smaller changes infw·
5~ 7
SENSITIVITY
COEFFICIENTS
The sensitivity analysis approach just presented can be developed further using the
properties of the logarithm and the variance operator. This can lead to some to some
interesting insights into the sources of variability and data error. This process is
sometimes called first-order or linearized error analysis.
Suppose we have an estimator of the form
where Y is some quantity to be estimated that depends on the values of the X 1, X2, ... ,
XJ, all independent quantities. We treat the Xi as random variables even though the
relationship is deterministic; this makes Y a random variable also. In general, the
119
Statistics for Petroleum Engineers and Geoscientists
function Y is known (usually it is a physical law) and is nonlinear. As discussed in
Chap. 1, statistics cannot help in determining the form of Y, but, once it is determined,
statistics can determine the sources of its variability. We could actually treat a system of
equations of the above form but will not for mathematical brevity.
As in Section 5-6, we linearize Y by first writing its differential expansion:
Upon multiplying and dividing each term on the right side by the respective Xi·
multiplying the entire expression by Y, and recognizing the differential properties of
natural logarithms, this equation becomes
d ln(Y) =ar d ln(X1) + a2 d ln(X2) + ··· + a1 d ln(XJ)
(5-1)
where the coefficients ai are sensitivity coefficients defmed as (Hirasaki, 1975)
These coefficients are not, at this point, constants since they depend on the values of the
entire set of Xi. The ai are the relative change in Y caused by a relative change in the
respective Xi. The relative part of this statement, which arises in the derivation because
of the use of the logarithms, is important because the Xi can be different from each other
by several factors of ten. Of course, since Y is known, the sensitivity coefficients can be
easily calculated.
Equation (5-l) is thus far without approximation. We now replace the differential
changes by discrete changes away from some base set of values (denoted by 0 ):
/
d ln(Xi)
0
0
=ln(Xi)- ln(Xi)
= ln(Xi I xi)
'
The resulting equation is now only an approximation, being dependent on the selection of
the base values. It is a good approximation only in some neighborhood of the base
values. If we, in addition, evaluate the ai at the base values, the estimator becomes linear
with constant coefficients.
f
e
If
e
IT
If
d
if
Chap. 5 Estimators and Their Assessment
120
I
ln(YIY0 )
=I
ai ln(Xi/Xf)
i=l
This linearized estimator is now in a suitable form for application of the variance
operator,
I
Var(Y)
=I a;
Var(Xi)
i=l
from the properties Var(aX) = a2Var(X), Var(X+ constant)= Var(X), and Var(constant) = 0.
The signs of the ai no longer matter since they appear squared. This expression assumes
that the Xi are independent.
Example 5- Sensitivities in the Carman-Kozeny Equation. We can use the
above procedure on a fairly simple equation to make inferences about the
origin of permeability heterogeneity. The application also illustrates other
ways to linearize estimators.
Let us view a reservoir as consisting of equal-sized patches, each composed of
spherical particles of constant diameter. (The facies-driven explanation of
heterogeneity discussed in Chap. 11 suggests that this picture is not too far
from reality, although we invoke it here mainly for mathematical
convenience.) The Carman-Kozeny (CK) equation
now gives the permeability within each patch. Dp is the particle diameter, (/>
the porosity, and r the tortuousity in this equation.
The CK equation is a mix of multiplications and subtractions. It can be
partially linearized simply by taking logarithms.
ln(k) = 2ln(Dp) -ln('r) + 1{(1 3
~ (/>)2] -ln(72)
Statistics for Petroleum Engineers and Geoscientists
121
From inspection, the sensitivity coefficients for Dp and-rare aDp = 2 and
a-r = -1. These coefficients are constant because of the logarithms. By
differentiating this expression, the sensitivity coefficient for porosity is
_l...::1!1!.
a(/J-
1 - (jJ
This requires a base value, which we take to be (jJ = 0.2 from which
at= 3.25. Note that the CK permeability estimate is most sensitive to
porosity, followed by particle size, then tortuousity (la¢1 > lanPI > la.tJ).
The variance ofln(k) is
I
Var[ln(k)] = ""'
.LJ ai2 Var(Xi) =at2 Var[ln(f)] +aD2 Var[ln(Dp)] +a-r2 Var[ln(r)]
~1
p
Estimating this variability requires estimates of the variabilities of ln(f),
ln(Dp). and ln( r). We note in passing that porosity, particle size, and
tortuousity are independent according to the patches model originally invoked.
Reasonable values for the variances are Var[ln((j))] =0.26, Var[ln(Dp)] =2.3,
and Var[ln(1')] = 0.69. These values are taken from typical core data. The
variance of ln(k) is now 13.08, but perhaps the most insight comes from
apportioning this variance among the terms. From this we find that 24% of
Var[ln(k)] comes from ln((j)), 70% from ln(Dp) and 6% from ln(t'). Substantially more of the variability in permeability comes from variability in
particle size, even though the estimate is more sensitive to porosity.
Additional calculations of this sort, that do not regard {j), Dp. and 1: as
independent, are to be found in Panda (1994).
Although simple, Example 5 illustrates some profound truths about the origin of
reservoir heterogeneity-namely that most heterogeneity arises because of particle size
variations. This observation accounts, in part, for the commonly poor quality of
permeability-porosity correlations. Similar analysis on other estimators could infer the
source of measurement errors. We conclude with the following three points.
1.
The procedure outlined above is the most general with respect to estimator
linearization. However, as the example shows, there are other means of
linearization, many of which do not involve approx,imation. These should be used
if the form of the estimator allows it. For example, logarithms are not needed if
Chap. 5 Estimators and Their Assessment
122
the estimator is originally linear. In such cases, the entire development should be
done with variables scaled to a standard normal distribution.
2.
The restriction to small changes away from a set of base values is unnecessarily
limiting in many cases. When the small change approximation is not acceptable,
the general nonlinear estimator Y= W(X1,X2, ... ,X]) can be used in Monte
Carlo algorithms that do not require the estimator to be linear (e.g., Tarantola,
1987). Many such algorithms also allow for the Xi to be dependent.
3. Perhaps most important is the fact that the procedure allows a distinction between
sensitivity and variability (or error). Sensitivity, as manifest through the
sensitivity coefficients, is merely a function of estimator form; variability is a
combination of sensitivity and the variability of quantities constituting the
estimator. In most engineering applications, variability is the most interesting
property.
5-8
CONFIDENCE INTERVALS
During estimation, the information conveyed by the data is "boiled down" by the
estimator W to produce the· estimate ~. There usually is information in the data that is
unused. Since~ is a random variable (e.g., Fig. 5-2), the additional information can be
used to estimate 'B's variability. This variability assessment, sf}.= [Var(~]l/2, provides a
confidence interval for ~.
Example 2 showed that estimator W 1 produced unbiased estimates with
/3/while W2 gave bias-corrected estimates with Var(~2) = 2fJ(I+2) for I
data. If we assume that the ~ 's are normally distributed, then ~~- N[ e, e2 !31] and
@1_ N[8, e2 JI(l+2)]. These results permit us to say, at any given level of probability,
how close our estimates @1 and ~2 are to the true value e (Fig. 5-4). For any number of
data and value of fJ, W2 will give estimates more closely centered about e.
Var(~l) =
rP
e
An alternative to showing estimated PDF's is to give the standard errors~. along with
the estimate, usually written as ~ ± s~. s~. is an estimate of [Var(~)]l/2 because we
(again) have to depend on the data. For example, in the case of W1 in Example 2a, we
have ~1 ± ( fJ 2 /3J)112. Clearly, if we are estimating(), we do not know() for the standard
error of ~1 either. We do the next best thing, however, and use the estimate ~1 again:
~1 ± ~13[)112 . In essence, we are giving our best estimate of ()along with our best
estimate of its standard deviation.
Statistics for Petroleum Engineers and Geoscientists
123
0.8
f(8')
0.6
0.4
0.2
0
-1
-2
Figure 5-4.
e,0
1
2
PDF's of ~1 and~ with thex axis scaled in units of (8!3!)112, centered at e.
We often assume that unbiased estimates are normally distributed. This is not always
true, particularly when I is small (e.g., less than 20), and theory exists for standard errors
of some estimators (e.g., the arithmetic average and the correlation coefficient). However,
as a working practice for I ~ 20, it is often safe to assume ~ is normally distributed. We
can then use the properties of the normal PDF to state how often 8 will be within a given
range of~- Hence, there is a 68% chance that(~- sf}. )< (8 < ~+sf}.) and a 95% chance
that (o(,e)- 2sf).)< (8 < ~ + 2sf).) (Fig. 5-5).
95%
~\
\\
0
I
A
e-2s~
9?
I
I
A
e-s~
I-cc
Figure 5-5.
~?I
A
e
68%
~?I 9?
I
A
S+s~
A
1»'-
e+2s~
~I
The true value for() is not known. The size of interval where it may be
located depends on the probability assignment.
124
Chap. 5 Estimators and Their Assessment
To use standard errors to define and interpret confidence·intervals requires, strictly
speaking, that the variable be ·additive (rati.Ometric, Chap. 3). For this type of variable,
we can add and subtract multiples of the standard error without difficulty. We do reach a
limit, however, because we might, for example, produce a reservoir thickness such as
h =22 m ± 15 m. The 95% confidence limits would be -8 m < h < 52 m, and we know
that lower extreme of -8. m cannot be correct This is because the assumption of a normal
PDF is no longer valid and we are seeing it break down. But there is also the more
difficult area where we want to use standard errors with interval-type variables, which may
notbe additive. These are very common in reservoir characterization (e.g., permeability,
density, anp resistivity).
The role of additivity in statistical ·analysis requires further research. Here, we will
take a pragmatic approach that does not demand unconditional additivity of the variables
to be analyzed. That is, if X is the reservoir property being considered, we will not
require that 'LaiXi be a physically meaningful quantity for all a's and X's. We will
only require "local" additivity, so that X± AX, where L1X is a "small" change in the
value X (say, IAXI :::;; 0.20X), is still physically meaningful. By assuming local
additivity, we can continue to interpret standard errors in the conventional way. Large
changes, however, may lead to nonphysical values. For example, porosity is clearly
nonadditive because it cannot exceed unity. It is, however, a ratio of additive quantities:
void volume Vv + total volume Vr, where Vv:::;; Vr. For Vr constant, a Vv
perturbation, AVv, is directly proportional to the change in porosity: At/)= AVv/Vy.
Hence, as long as Vv + AVv:::;; Vr and Vr is constant, porosity is locally additive. If
porosities with differentmeasurement volumes are being added, they can be put on a
comparable basis by adjusting for Vy.
The traditional definition of confidence intervals is tied to Var~. Confidence intervals
are also just one type of statistical interval that could be considered. Others exist and are
discussed at length in Hahn and Meeker (1991). We will confine discussion here to the
traditional usage.
5·9
COMPUTATIONAL METHODS TO OBTAIN
CONFIDENCE INTERVALS
While a very useful tool for conveying uncertainty, confidence intervals in many
situations may be difficult to obtain or inaccurate, depending upon the exact
circumstances of the problem. In particular, small data sets from non-normal populations
can give rise to biased estimates and produce erroneous confidence regions if the usual
normal-theory assessments are used. Two procedures, called the jackknife and the
bootstrap, address these problems. The idea of these methods is to assess the variability
of estimates using incomplete data sets. That is, if we have I samples in a data set, we
125
Statistics for Petroleum Engineers and Geoscientists
can generate a number of smaller-size data sets from it. The jackknife and bootstrap
methods then use these subsets to assess the variability of the estimate.
Consider, for example, the jackknife technique on a data set of size I using an
estimator W. It is possible to generate I subsets of size I -1 by dropping a different datum
for each subset. Then, using W, we can produce an estimate for each of the I subsets.
Let @i represent the estimate based on the ith subset, i = 1, 2, ... ,I and let @represent the
estimate based on the full set of size I. An unbiased estimate of 8 is given by
I- 1 I
B*=I~ ---I,Bi
I
i=l
A variance estimate for 8*, which we need to obtain a confidence interval, is given by
S
I II( ~.
2 - ---~
8*-
I
I)
~· 2
-1 ~
L.
,-I..(,.,~
. 1
I=
J=l
'1
We can then apply the usual confidence interval techniques previously discussed to
indicate the variability in 8*. Clearly, if I exceeds 5 or 10, the jackknife becomes
cumbersome, making it a perfect task for a computer. Some of the newer data-analysis
packages have this feature available.
The bootstrap is even more computationally intensive than the jackknife but similar in
approach. The bootstrap generates subsets allowing replication of data, however. Hence,
a much larger number of subsets can be generated. As for the jackknife, W is used to
produce an estimate ~j for j = 1, 2, ... ,1, where J usually exceeds I. The estimate 1J for
the complete data set is given by
J
~-l"
- ]L. 11·'}
j=l
with a variance of
J
s2 =-1~ (~·
tJ J-lL. J
j=l
_ §)2
126
Chap. 5 Estimators and Their Assessment
The jackknife or bootstrap are useful not only for obtaining confidence intervals of X.
For example, these methods may be very useful for analyzing output from stochastic
simulations, where each run can be very costly and, hence, there are very few runs to
analyze. Many of the results are likely to come from non-normal distributions (e.g.,
produced quantities, time to breakthrough, and water cut). Hence, the "average"
performance may be hard to assess without the jackknife or bootstrap. Similar arguments
can apply to geologic analyses, too. For example, calculation of R2 estimates from
small numbers of core-plug porosity and permeability data may be analyzed using the
jackknife or bootstrap. The reader is referred to Schiffelbein (1987) and Lewis and Orav
(1989) for further details and references.
Example 6- Jackknifing the Median. We compute the jackknife estimate of
the median from ten wireline porosity measurements, X1, ... ,X1Q, shown in
Table 5-l. ko.so = (0.191 + 0.206)/2 = 0.199. ko.so,i is the sample
median of the data set without the ith point in it.
Table 5-1. Example 5 jackknife analysis.
i
Xi
0.114
0.146
0.178
0.181
0.191
0.206
0.207
0.208
0.218
0.242
1
2
3
4
5
6
7
8
9
10
j( 0.50 i
0.206
0.206
0.206
0.206
0.206
0.191
0.191
0.191
0.191
0.191
Figure 5-6 is a probability plot of the data, suggesting they appear to come
from a normal PDF. Because the data appear to have a normal PDF,ko.so
is unbiased. Hence, we do not expect the jackknife to produce an estimate
different from ko.so and that is the case:
M
.fl.
/-1
x 050 = Ixo.so- 1
I
~ A
~
xo.so,i
9
= 10•0.199- 10 (5 • 0.206 + 5 • 0.191)
i=l
= 0.199
127
Statistics for Petroleum Engineers and Geoscientists
Note that we are maintaining three-figure accuracy, in keeping with the
original data.
Porosity
0.25
*
** *
*
*
**
*
*
-3
-2
-1
0
2
1
3
Normal Qu.antiles
Figure 5-6.
Porosity data probability plot showing approximately normal PDF.
The jackknife, however, also provides a confidence interval estimate of the sample
median:
sg*
0.50
I- 1 I (
-I,
=1
1 I .
to.so,i- f.L
J= 1
zo.50,j
)2
i=l
= { 0 [5 (0.206- o.I99) 2+ s (0.191 - o.I99)2J
= 0.000509
Chap. 5 Estimators and Their Assessment
.128
sx*.0.50 = 0.0225
.
or
Thus, at 95% probability, the confidence interval for k~.so is
from(k~.so - 2s .* ) to(k~.so + 2s ·• ), or 0.154 < k~.so < 0.244.
Xo.so
Xo.so
Clearly, there is a large possible range for k~.so· Theoretical results,
assuming Xi is normally distributed, give s •• =0~020 (Kendall and Stuart,
Xo.so
1977, p. 351), which agrees well with the jackknife value.
The jackknife procedure is not restricted to subdividing I data into subsets of size I-1. The
procedure is more general. Lewis and Orav (1989, Chap. 9) have the details.
5-10 PROPERTIES OF THE SAMPLE MEAN
The arithmetic average is a very common measure of central tendency. It has been
extensively studied and we give some results here. Let Xi be a random sample of I
observations from a distribution for which the population mean J.L and variance CY2 exist.
The arithmetic average or sample mean·is
- 1 I
X=- LXi
I i=l
F~m
the Central Limit Theorem (Chap. 4), one can show (Cramer, 1946, Chap. 28) that
X has the following properties:
(J'2
X - N(J.L, / ) for large I
~
~
X- N(Jl, / ) exactly, if Xi= N(Jl, 0'":')
where E(Xi) = J.L and Var(Xi) = 0'2. The first result is independent of the PDF of the
random variable; it says that X is an unbiased estimate of J.L or, on avera~. X over all
samples is equal to the "true" sample space mean. In some cases, X approaches
normality for surprisingly small I. Papoulis (1965, pp. 267-268) gives an example of
independent, uniformly distributed X and I= 3.
Statistics for Petroleum Engineers and Geoscientists
129
One of the interesting features of the arithmetic a~erage is how its variability.depends
on the number of samples, I. The variability of X decreases as I increases, which is
what we would expect; the more information we put in, the better the estimate ought to
be. In particular, the standard deviation of X is proportional to I-1/2, a property
common to many estimators (e.g., Example 2a).
Before we consider further the sampling properties of X, we first discuss variance
estimation.
5-11 PROPERTIES OF THE SAMPLE VARIANCE
The sample variance is a measure of dispersion. It is defined as
t~
I
s2 =
(Xi - x)2
i=l
s2 is biased since we can show (Meyer, 1966, p. 254) that
1
2 =IE(s)
--a 2
I
However, just as in Example 2b, we can formulate an unbiased estimate of a 2 by using
AsL_I_s2
-I- 1
I
= -1- " (Xi-
I- 1 ~
X) 2
i=l
This result has a more general concept embedded in the use of (I- 1) instead of I in the
denominator. Since the sample mean is required to compute the sample variance, we have
effectively reduced the number of independent pieces of information about the sample
variance by one, leaving (I- 1) "degrees of freedom." Therefore, (I- 1) is a "natural"
denominator for a sum of squared deviations from the sample mean. Using (I- 1) instead
of I to estimate the variance is not very important for large samples, but we should
always use it for small data sets (/ < 20).
Chap. 5 Estimators and Their Assessment
130
If X- N(Jl, a2), then the Var(~2) = a2/21 (Kendall and Stuart, 1977, p. 249). More
generally, the variability of~2 is approximately
r"2)
Var,s
=
E {[X - E (X)] 4 } - Var(X)2
4(/- l)Var(X)
where E {[X- E(X)]4} is the fourth centered moment of X. This result follows the general
pattern that the estimate variability for the kfh moment depends upon the 2kth moment of
the PDF.
To estimate the standard deviation, one may usually take the square root of the
variance. For very small data sets (/ ~ 5), however, another effect comes in. The square
root is a nonlinear function and causes the estimated standard deviation to be too small.
When I~ 5, multiply ~by 1+ (1/4/) (Johnson and Kotz, 1970, pp. 62-63).
5~12
CONFIDENCE INTERVALS FOR THE SAMPLE
MEAN
The arithmetic average, X, is often useQ. to estimate E(X). There are some theoretical
results for the confidence limits of X when the samples come from a normally
distributed population. If we know the population variance, a2 = Var(X), we would
know that (Jl-zat{D ~ X~ (Jl+zat{D for a fraction [1 - P(z)]/2 of the time. (Recall P
is the standard normal CDF.) Usually, howe~r, we do not know a but can estimate it
from the same samples we used to calculate X. If that is the case, we have to introduce
a different term that acknowledges that~ is also subject to statistical variation:
J1 =
x ± t(a/2, df> ~r[i
where t(a./2, df) is the "t value" from Student's distribution (Fig. 5-7) with confidence
level a, df =I - 1 degrees of freedom, and ~is the sample standard deviation.
Values of the function t(a/2, df) are widely tabulated (e.g., Table 5-2). a is the
complement of the fractional confidence limit (e.g., if we want 95% confidence limits,
then a= 0.05). As df (i.e., the sample size) becomes large, the t value approaches the
normal PDF value for the same confidence level. For a given value of t = to, the
symmetrical form of Student's distribution implies that any random variable T with this
PDF is just as likely to exceed some value to as it is to be less than -to. Consequently,
when we set a confidence level a forT, we usually want t(a/2, df) because T can be
either positive or negative with equal probability. For example, if our confidence level is
(l - a) = 0.95, we find t(0.025, df). This situation is called a double-sided confidence
Statistics for Petroleum Engineers and Geoscientists
131
interval. If we knew, however, that T could only have one sign or we were interested
only in excursions in one direction, then we would use t(a, df) for a single-sided
confidence interval.
Table 5-2. t values.
df
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
18
20
22
24
26
28
30
40
60
Tail area :probability, a
0.05
0.025
0.005
6.314
12.706
63.657
2.920
4.303
9.925
2.353
3.182
5.841
2.132
2.776
4.604
2.571
2.015
4.032
1.943
2.447
3.707
2.365
1.895
3.499
2.306
1.860
3.355
1.833
2.262
3.250
1.812
2.228
3.169
2.201
1.796
3.106
1.782
2.179
3.055
1.771
2.160
3.012
1.761
2.145
2.977
2.131
1.753
2.947
1.746
2.120
2.921
1.734
2.101
2.878
1.725
2.086
2.845
1.717
2.074
2.819
2.064
1.711
2.797
2.056
1.706
2.779
1.701
2.048
2.763
1.697
2.042
2.750
1.684
2.021
2.704
1.671
2.000
2.660
Chap. 5 Estimators and Their Assessment
132
f
df~oo
t
0
Figure 5-7.
The Student's distributions change shape with the degrees of freedom (df).
t is the value for which the tail area is a at the stipulated df value.
Example 7- Estimating the Mean Value. Suppose we wish to specify the
95% (a= 0.05) confidence interval for the population mean from a sample of
25 data points (df 24) with X= 30and 1 3. The t value from the above
table is 2.064 and the above formula yields
=
=
Jl = 30 ± (2.064) 3r/2s
or
Jl
E
{28.76, 31.23}
Thus, we can be 95% certain that the population mean is within the stated
limits (Fig. 5-8).
~~.------95%------~
..~
~?
~?
~?
~----~~~,--+1~1------~----~~~--~~1
0
Figure 5-8.
28.76
30.00
31.23
Confidence limits for X. There is a 2.5% chance that Jl >
31.23 and a similar chance that Jl < 28.76.
133
Statistics for Petroleum Engineers and Geoscientists
Had we used the z value instead of the t value (i.e., if we ignored the fact that
we have estimated the standard deviation from the data as well as the mean),
the factor would be 1.960 (from normal table, Chap. 4) instead of 2.064.
Despite its considerable attractions, there are some problems with the sample mean. It
can be inefficient when the Xi are coming from skewed populations, such as the lognormal PDF. Consequently, we can expect that the arithmetic averages of quantities like
permeability and grain size may be inaccurate. Agterberg (1974, pp. 235ft) discusses an
efficient method for estimating the mean for log-normal populations.
5·13 PROPERTIES OF THE SAMPLE MEDIAN
Besides the sample mean, the ~ample median io.so is another measure of central
tendency. We have already observed thatio.so is more robust than X. io.so.
however, doe~ not have the nice distributional properties obtained through the central
limit effect that X has.
The exact PDF and properties of io.so are discussed by Kendall and Stuart (1977,
pp. 252, 348-351). When X- N(J.L, u2), io.so is unbiased and its standard error is
approximated by
""var(io.so) =
for I data. Since Var(
X)= u2ti
we find that, for large/,
Var( X)
Var(io.so)
~x
~
u2jJ = 2/n
ndl-!21
estimatio~
Hence,
=io.SO· So, for the normal PDF, where Xo.so =E(X),
of
the mean using i"o.so is less efficient (i.e., needs about 60% more data) than using X.
5·14 PROPERTIES OF THE INTERQUARTILE RANGE
The interquartile range (IQR) is the difference (X0.75- Xo.25) and is another measure of
dispersion. It has been used as a robust substitute for the standard deviation, .Vvar(X).
Apart from robustness, the sample IQR, (i o. 75 - io.25),. has several other features in
common with the median. Both are based on order statistics, which are sample quantiles.
Chap. 5 Estimators and Their Assessment
134
For X - N(/1., cil), (Xo.75 - Xo.25) = 1.35cr. Hence, .besides the sample variance, an
estimator for cr is ~QR = (io:is - io.25)/1.35 when Xis normally distributed. For
large I, the sample IQR is unbiased and has the amp ling variability
Var(i" o. 7 5 - i"o.25) =3 a2 /2/ (Kendall and Stuart, 1977, p. 350). ~QR is less efficient
than~ at estimating cr2. Depending upon/, Var(~QR) is between 30% (/ =6) and 65%
(/large) higher than Var(s'2). This puts ~QR at about the same efficiency as the sample
median for estimating their respective quantities when X is normally distributed.
5-15 ESTIMATOR FOR NONARITHMETIC MEANS
There are other meansbesides the arithmetic mean, XA = E(X}, that arise in reservoir
evaluation. The geometric and harmonic means of X are defined as Xo =exp(E[ln(X)]}
andXH = [E(X-l}rl, respectively. All three means are specific instances of a more
general situation (Kendall and Stuart, 1977, pp. 37-38), the power mean:
[E(XP)]l/p
{
Xp =
exp(E[ln(X)]}
p
*· 0
p
=0
XH is the case when p = -1, X a when p = 0, and XA corresponds top= 1. Xp can be
viewed as being the arithmetic mean of xP raised to the 1/p power. As long as the
reservoir property X is nonnegative, Xp 1 ~ Xp 2 if Pl ~ P2 (Kendall and Stuart,
1977, p. 38). Hence, XH :5.Xo :5.XA for many reservoir properties.
We have already considered estimation of XA. Compared to XA, the properties of Xo
and XH estimators have received little attention. Two common estimators for X a and
XHare
7.£ ln(Xi)]
ll=l
Xo =exJ
(geometric average)
and
(harmonic average)
Statistics for Petroleum Engineers and Geoscientists
135
Neither of these is unbiased. For example, Xa overestimates X a if ln(X)- N(Jl., o-2):
E( Xa) = Xa exp(o- 2/2!), so Xa is consistent. Under similar conditions,
E( XH) = XH{l + [exp(o-2)- 1]/I}. Xa and XH are not especially robust, either.
These two estimators are sensitive to small, uncertain data values. One way to avoid this
problem is to use the sample median: If Y =XP, chooseip = rto.soJ 11P. Another
approach is to use a trimmed or Winsorized mean of Y, which is discussed below.
5-16 FLOW ASSOCIATIONS FOR MEAN VALUES
As we pointed out at the beginning of this chapter, an important element of the
estimation procedure is to decide on the quantity to be estimated, 8. Sometimes the
decision is fairly obvious. For example, E(cp) is often a useful measure for porosity in
many applications because it represents a central value for a physically interpretable
quantity. Permeability, on the other hand, is usually a more difficult property for which
to find a representative value, and only a few systematic guidelines are available.
Why is permeability so troublesome? Since it pertains to flow, permeability is clearly
an important property. But, permeability is an intensive variable (Chap. 3). Depending
upon the situation, it may not be additive. The flow transmitted by any given region
depends upon the permeabilities of surrounding regions; For example, consider a highly
permeable region of a reservoir encased in shale (e.g., lenticular bedding). No matter how
permeable is the center, the outer "shell" of low-permeability material prevents flow.
This is not the case for porosity, for example, where the pore volume of fluids in a lens
contributes to the total amount of fluid in a_region without regard to whether the fluids
can move or not. There are a few cases, however, where it is clear what the additive
properties of permeability are.
In the case of linear flow parallel to a stratified medium, typical of shallow marine
sheet sands, of I layers (Fig. 5-9), the aggregate permeability (kr) of the region is the
expected value of the layer permeabilities. To see this, we take the expression for k1 and
rearrange it:
I
L. kihi
i=l
kt=
I
L. hi
i=l
I
= ~)i
i=l
hi
I·
-I--= L ki Pi= E(k)
hj
i=l
j=l
l
Chap. 5 Estimators and Their Assessment
136
since Pi is the probability that the permeability is ki- Clearly, permeability is additive in
this case.
kl
FLOW
•
h .4l
1.,,
.4~
kr
Figure 5-9.
hr,,
Effective permeability for layer-parallel flow.
When linear flow is orthogonal to the layers (e.g., dune crossbeds), permeability is no
longer additive. Its inverse, however, is additive. The aggregate resistance to flow,
k-/.
is given by the expected value of the layer resistances to flow (Fig. 5-10):
I
L
-1
kt
hi/ki
i=l
I
=
I
I
1
= Lki
hi
i=l
i=l
Consequently, here are two situations for which we know the quantity to be estimated,
8, when the aggregate permeability is desired:
kt =E(k)
k-/
=E(k- 1)
for linear flow parallel to layers
for linear flow normal to layers
Statistics for Petroleum Engineers and Geoscientists
kl
137
h ~t.
1..,,
FLOWt
.h.
ki
Figure 5-10.
hI
-,r
Effective permeability for layer-transverse flow.
Many geological systems, while containing some degree of ordered permeability
variation, do not meet the layered situations described above (e.g., turbidites, braided
deposits). What are the expressions for kt in those cases? The best general answer comes
from Matheron (1967): E(k) ~ kt ~ [E(k-1 )]-1. In other words, kt appears to be the
power mean of permeability for some exponent p, -1 :5: p :5: 1. The work of Desbarats
(1987) has supported this result, but it is not clear how to select p a priori.
There is one other situation for which theoretical results exist (Matheron, 1967).
When ln(k)-N(Jl, cr2), a is small, and the flow is two-dimensional (2D),
ln(kt) = E[ln(k)]. Hence, the geometric mean is appropriate. Various simulation studies
(e.g., Warren and Price, 1961) have extended this result to less-restrictive situations with
some success. This result does not require that the system be totally disordered. As long
as the scale of the ordered or structured element is small compared to the region for which
kt is sought, the geometric mean will apply, given that k is log-normally distributed and
exhibits only moderate variation.
In summary, then, there are three definite results-all assuming linear 2D flow-for
determining kt.
1.
Layered system, flow parallel to layers: arithmetic mean.
2.
Layered system, flow normal to layers: harmonic mean.
3.
Random system, log-normal k, small variation: geometric mean.
Otherwise, E(k) ~ kt ~ [E(k-1)]-1 is the best that can be said. The three definitive results
suggest that some function of permeability may always be additive, but the precise form
of the function changes with the flow geometry.
Chap. 5 Estimators and Their Assessment
138
5a17 PHYSICAL INTERPRETABILITY IN ESTIMATION
Any estimator will manipulate data in some way to produce an estimate. Despite the
large amount written on estimation, it appears very few have considered whether the
estimator combines variable values in a physically meaningful manner. For example, to
obtain an arithmetic average, sample values that are physically meaningful have to be
added together. To add values and produce something that is still physically meaningful
requires that the variable be ratiometric (Chap. 3), i.e., additive. We have just observed
that permeability may not be additive, so calculating the average permeability may give a
nonsensical result.
A still more questionable estimator is the sample variance. Differences of sample
values are taken, squared, and summed. This suggests that the variable and its square are
both additive, a situation that would apply, for example, with length because area is also
additive. In this light, the sample IQR might be preferable because it does not take
squares. Unfortunately, we cannot simply take the local additivity approach with
estimators that we took with their standard errors. Some estimators, such as the
arithmetic average or sample variance, add or subtract quantities of similar magnitude and
manipulate them further. It is not clear what effect the lack of additivity has on results
and further research is needed.
In this respect, PDF's could play an interesting role. It is well-known that two
variables with normal PDF's give a normally distributed sum (Cramer, 1946, p. 212).
The squares of normal PDF variables are also additive, giving the %-squared PDF
(Cramer, 1946, p. 233). Hence, statistical additivity (preservation of the PDF when
variables are added or subtracted) is satisfied for the average or variance when normal PDF
variables are involved. It may be that the effects of physical nonadditivity are mitigated
when variables have normal PDF's prior to use in estimators. For example, for sound
statistical reasons, Xo.so ::/:. XA when X is log-normally distributed (Chap. 4).
If ln(X)- N(J.l, o-2) and cr > 0, then Xo.so
ell-+ 0.5a2 .
0.5 =
= eJL
compared to E(X)
= XA
=
Xo.so does not involve a summation of X (recall that Xo.so is defined as
f f(x)dx),
but only summation of X's PDF, f(x). Compare this with XA that
f
does involve a summation of all possible values of X: XA = xf(x)dx. Could it be that
the difference between XA and Xo.so is also a reflection of X's nonadditivity?
Until this issue is resolved, it would appear that two approaches might be taken:
(1) use the median and IQR instead of the mean and standard deviation for central
tendency, dispersion, and variability measures, or (2) continue to use both the arithmetic
average and standard deviation, but also calculate the sample median and IQR to compare
and diagnose nonadditivity. For example, use ( XA- ko.so) ::/:. 0 as a diagnostic tool.
When variable PDF's are near-normal, the median and mean are similar in value; they can
Statistics for Petroleum Engineers and Geoscientists
139
differ substantially for non-normal PDF's. In the latter case, transform the variable to
normal (see Chap. 4) before applying estimators or re-express (if possible) the variable in
terms of quantities that are additive.
5~18
WEIGHTED
ESTIMATORS
Every estimator implicitly weights each data point upon which it operates. XA, for
example, weights all the data equally. Many other estimators give equal influence or
weight to the data points. This need not necessarily be the case, however.
There are several reasons for weighting some data more heavily than others. First,
some data may be more prone to errors than others. Particularly small or large values
may be poorly measured, and we do not want especially inaccurate data to have as much
influence as the more precisely measured values. Second, all the data may not be equally
representative. In Fig. 5-11, for example, a representative reservoir porosity would
4
probably not be given
by~ I, r/Ji
because ¢1, ¢2, and ¢3 represent similar portions of
i=l
the reservoir, the crest, while ¢4 represents only a portion of the flank-region porosity.
In other words, the four measurements do not represent equal amounts of information
concerning the large-scale reservoir porosity. Sampling points are seldom chosen at
random. Third, the data may represent different portions of the whole. Measurements
may be made on different volumes of rock and, therefore, need a weight that reflects the
amount of material sampled.
Figure 5-11. Porosity measurements from four locations in a reservoir.
Chap. 5 Estimators and Their Assessment
140
If W(XI.X2, ... ,XJ) is an estimator, then W(X1.X2, ... ,XJ, a1, a2, ... ,a1) is the
I
weighted estimator, where the ds are the weights and I,ai = 1 is a condition to ensure
i=l
unbiasedness. For example, if XA = W, then
by (Hald, 1952, pp. 243-245)
x,t is the weighted sample mean given
Weighted averages occur in the Kriging estimator to be discussed in Chap. 12. The
weighted sample variance is
When a1 = a2 = ···= a1 = 1, we have the unweighted version. If we further assume
that X 1 :::;, X2 :::;, · · ·:::;, XJ, then letting a1 = a2 = ···= ar = a(l-r+l) = a(l-r+2) =···=
a[= 0 and a(r+l) = a(r+2) = ···= a(l-r) =I!(I- 2r) gives the r-fold symmetrically
trimmed estimator. In other words, the trimmed estimator ignores the r lowest and r
highest values in the data set. A variation on trimming is to Winsorize the data set:
a1 = a2 =···=a,= a(/-r+l) = a(l-r+2) = ···= a:r = 0 (as with trimming), but a(r+l) =
a(l-r) = r + 1 and a(r+2) = · · · = a(I-r-1) = 1. Winsorizing still ignores the r lowest and
highest data, but it replaces them with r values of Xr and X(l-r)· Asymmetrical trimming
is also possible, but it is usually reserved for cases where the PDF is asymmetrical.
Barnett and Lewis (1984, Chap. 3) discuss these methods further.
Trimming or Winsorizing is useful when data sets are censored and the proportions of
lost data are known or when some data at the extremes are particularly corrupted. If the
variances of the corrupting factors are known, the weights can be modified so that all data
are included. For example, suppose data points 1, 2, ... , r are obtained using an
instrument with normal measurement errors of variance
a;
while r+l, r+2, ... ,
2
. b'l'
. h ts IS
.
varm
1 1ty a 2. A set o f we1g
al
= a2 =
... = a, =
2
2
ra 2 + (n-r)Ci 1
n
have
Statistics for Petroleum Engineers and Geoscientists
141
and
a(r+l) = a(r+2) = ··· = a1 =
2
2
rCJ2 + (n-r)CJ 1
More sophisticated approaches are available, depending upon the information available
about the measurement properties (Fuller, 1987).
5-19
SUMMARY
REMARKS
Estimation is an important aspect of data analysis. It is important to understand what the
estimator properties are so that the most appropriate estimator is used for the analysis at
hand. The uncertainties of estimates can be assessed using confidence intervals.
Confidence-interval size is a strong function of data variability; a highly variable
material produces estimates with large confidence intervals. Since the geological medium
has variability that cannot be changed, the only methods for reducing the size of
confidence intervals is to collect more data or apply an estimator that uses the data more
efficiently. Chapter 6 presents a method for gauging the number of samples needed, based
on the geological characterization ·and heterogeneity of the rock.
6
F
EITY
The most common statistics of a random variable are measures of central tendency,
dispersion, and correlation. Chapter 5 dealt with measures of central tendency and their
associated uses, along with introducing some simple measures of dispersion. This
chapter further discusses measures of dispersion or variation with a special emphasis on
measures used in assessing their impact on flow performance.
A measure of variability can, of course, be applied to any reservoir property.
However, permeability varies far more than other properties that affect flow and
displacement. Hence, in the petroleum sciences, measures of heterogeneity are almost
exclusively applied to penneability data.
Heterogeneity measures are useful for a number of purposes. Since heterogeneity
influences the performance of many flow processes, it is helpful to have a single statistic
that will convey the permeability variation. Variabilities can be compared for
geologically similar units and sampling schemes can be adjusted for the variability
present. Performance models have been developed that show how permeability
heterogeneity will influence a particular recovery process (e.g., Lake, 1989, pp. 411-416).
Heterogeneity measures are also helpful when comparing performance for two or more
fields. It should be kept in mind, however, that in summarizing variability, a
considerable amount of information is lost.
143
Chap. 6
144
Measures of Heterogeneity
Heterogeneity measures are not a substitute for detailed geological study,
measurements, and reservoir analysis. They are simply one way of beginning to assess a
formation unit. Most measures do not include any information about spatial arrangement
and, even when they do, they tend to ignore most of the structure present.
6-1
DEFINITION OF HETEROGENEITY
Informally, we use heterogeneity and variability interchangeably. In reservoir
characterization, however, heterogeneity specifically applies to variability that affects.
flow. Consider a high-rate displacement using matched mobility and density and
chemically inert, miscible fluids. Heterogeneity is the property of the medium that
causes the flood front, the boundary between the displacing and displaced fluids, to distort
and spread as the displacement proceeds. Permeability variation is usually the prime
cause of flood-front spread and distortion; for a displacement in a hypothetical
homogeneous medium, the rate of distortion and spreading is zero. As the permeability
variability increases, both distortion and spreading increase. Of course, the arrangement
of the permeability is also important since this governs the number and size of interwell
pathways. We discuss ways to measure spatial arrangement in Chap. 11.
Whatever the reservoir properties involved, heterogeneity measures can be classified
into two groups, static and dynamic. Static measures are based on measured samples
from the formation and require some flow model to be used to interpret the effect of
variability on flow. Dynamic measures use a flow experiment and are, therefore, a direct
measure of how the heterogeneity affects the flow.
Each measure type has advantages and disadvantages. For example, an advantage to
dynamic measures is that, if the process used during the flow experiment closely parallels
the process that is expected to be applied to the reservoir, the results are most directly
applicable with a minimum of interpretation. Disadvantages include the cost, the
complexity, and the selection of "representative" elements of the reservoir for conducting
the flow experiments at the appropriate scale.
6-2
STATIC
MEASURES
We discuss four types of static heterogeneity measures and a few of their properties.
The Coefficient of Variation
A static measure often used in describing the amount of variation in a population is the
coefficient of variation,
Statistics for Petroleum Engineers and Geoscientists
145
cV--~
~'
E(k)
This dimensionless measure of sample variability or dispersion, introduced in Chap. 4,
expresses the standard deviation as a fraction of the m~an. For data from different
popl.liiililinffor sorn:ces;tfie mean and standard deviation often tend to change together such
that Cy remains relatively constant. Any large changes in Cy between two samples
would indicate a dramatic difference in the populations associated with those samples.
1\
1\
-
A Cv estimator, based on the sample mean and standard deviation, is Cy = s lkAThe statistical properties of "Cy are not easily determined in general. Hald (1952, pp.
301-302) gives results for samples from a normal population and Koopmans et al. (1964)
give results for the log-normal case. Figure 6-1 shows these results for a sample size
I= 25. For Cv :::; 0.5, the sampling variabilities from the two distributions are quite
similar. When Cy > 0.5, where a normal PDF is not possible for a nonnegative
variable, the lower and upper limits become increasingly asymmetrical about the line
"
"
.I-Cy= Cy. For I samples, Cy:::; 'I I- 1 (Kendall and Stuart, 1977, p. 48), which is
achieved when (I -1) samples are one value and one sample is another value.
1\
-
Estimators other than s lkA may be used, if the PDF is known, with some increase in
efficiency. For example, if ln(k)- N(/1. a2), kAikH = exp(a2), where kA and kH are the
arithmetic and harmonic means (Johnson and Kotz, 1970, p. 115). Hence,
The coefficient of variation is being increasingly applied in geological andengineering
~stud1es as an asses~-~~ii.~~~P§!.~~~!)j}}tyheterog~n~f!Y~ Cv has been used in a·sil:idy-of
the effects of heterogeneity and structure upon unstable miscible displacements (Moissis
and Wheeler, 1990). It is also useful when comparing variabilities of different facies,
particularly when there can be competing causes for permeability variation. Corbett and
Jensen (1991) for example, used Cy to assess the relative effects of grain-size variation
and mica content upon permeability variation. Comparisons of geologically similar
elements in outcrop and subsurface showed that, despite large changes in the average
permeability, the Cy's remained similar (Goggin et al., 1992; Kittridge et al., 1990).
These and other studies (e.g., Corbett and Jensen, 1992a) suggest that Cy's may be
transportable for elements with similar geologies. Cy's and scale may also be linked as
shown in the following example.
Chap. 6
146
Measures of Heterogeneity
5.00
4.00
3.00
2.00
1.00
0.00 -4ll!lll!ll:;;----,i:...;,.;,;.=;;;;..;=;,;:.;;..----.-------.
2.0
1.0
1.5
0.0
0.5
Cv
Figure 6-1.
A
Cy variability (J =25 samples) for normal and log-normal populations.
Example 1- Geological Variation and Scale. Cy, as a measure of relative
permeability variation, can be used to compare and contrast facies and
formations. The smaller stratal elements appear to have the most uniform
properties. The larger the element, the more opportunity for variability
(Fig. 6-2).
In Fig. 6-2, we have not highlighted the scale (i.e., grid size) associated with
each measurement. We prefer to note lamina or subfacies (e.g., grainfall,
grainflow), bedform or facies (e.g., trough crossbedding), or formation (e.g.,
Etive, Rannoch, Rotliegendes) terminology in preference to specific
dimensions. If a bedform consists of a single lamina type, or a formation
consists of a single bedform type, then it is reasonable to expect similar
variabilities with increasing scale (e.g., low-contrast lamination and SCS,
HCS, and Rannoch). The occurrence of a rock type with a certain Cy (e.g.,
fluvial trough crossbeds) does not mean that they always occur with this level
of heterogeneity, as has been pointed out by Martinus and Nieuwenhuijs
147
Statistics for Petroleum Engineers and Geoscientists
(1995). The interesting point to note in Fig. 6-2 is that the two samples of
fluvial trough crossbeds (from the same formation) have similar Cy's and,
measured by different geologists, this suggests a certain transportability.
With the combination of several subfacies or bedforms, the variability
increases. For example, bedforms are composed of numerous laminations. In
the aeolian environment, dune bedforms are composed of wind-ripple and
grainflow components, each of which has an intrinsic variability. Goggin et
al. (1988) found that Cy= 0.26 for wind ripple (unimodal PDF) and 0.21 for
grainflow (unimodal PDF) in a unit of the Page Sandstone. The combination
for the dune (bimodal PDF) produced Cy = 0.45, approximately the sum of
the grainflow and wind-ripple Cy's.
Carbonate (mixed pore type) (4)
S.North Sea Rotliegendes Fm (6)
Crevasse splay sst (5)
Shallow marine rippled micaceous sst
Fluvial lateral accretion sst (5)
Distributary/tidal channel Etive ssts
Beach/stacked tidal Etive Fm.
Heterolithic channel fill
Shallow marine HCS
Shallow marine high contrast lamination
Shallow marine Lochaline Sst (3)
Shallow marine Rannach Fm
Aeolian interdune (1)
Shallow marine SCS
Large scale cross-bed channel (5)
Mixed aeolian wind ripple/grainflow(1)
Fluvial trough-cross beds (5)
Fluvial trough-cross beds (2)
Shallow marine low contrast lamination
Aeolian grainflow (1)
Aeolian wind ripple (1)
Homogeneous core plugs
Synthetic core plugs
Heterogeneous
Homogeneous
0
1
2
3
4
Cv
Figure 6-2. Variation of Cy with scale and depositional environment. SCS =
swaley cross-stratification and HCS = hummocky cross-stratification.
Homogeneous region is taken as Cy s; 0.5; 0.5 < Cv s; 1 is
heterogeneous; and Cy > 1 is very heterogeneous. Sources .of data for
this plot are (1) Goggin et al, (1988); (2) Dreyer et al. (1990); (3)
Lewis and Lowden (1990); (4) Kittridge et al. (1990)~ (5) Jacobsen and
Rendall (1991); and (6) Rosvoll, personal communication (1991).
Chap. 6
148
Measures of Heterogeneity
Corbett and Jensen (1993b) observed that an HCS bedform in the shallow
marine environment has the following facies variabilities: Cv =0.2 to 0.7
for the low-contrast (low-mica) laminations; Cv = 1.0 to 1.3 for the highcontrast (high-mica) laminations; and Cv = 1.0 to 1.3 for the rippled
laminations. The HCS bedform is approximately composed of 60% lowcontrast rock, 35% high-contrast rock, and 5% rippled material. The weighted
sum of the component Cv's gives 0.60 • 0.6 + 0.35 • 1.2 + 0.05 • 1.2 =
0.84. This agrees well with the overall HCS Cv of 0.86.
Despite the evidence of this example, Cv's are not, in general, additive. For example,
consider a formation unit composed of two elements (a mixture), each having independent
random permeabilities, k1 and k2 , and present in proportions p and 1-p, respectively. For
this mixture of random variables, let E(k1) = fJ.l• Var(kl)
= ai, E(k2) = fJ.2,
and
Var(k2) = a;. The Cv of the unit is given by (Chap. 4, Example 4)
Cv
-..J PaT + (l-p
-
)a;
+ p(l-p )(fJ.l - J1.2) 2
PfJ.l + (l-p)JL 2
total -
which is not the sum of the constituent Cv's. If we let JL 2 = rJL 1 for some factor r,
Cvtotal can be expressed as
cvtota/ --
~ pC~ + (l-p)r2c~2 + p(l-p)(l-r) 2
1
p + r(l-p)
When the component means are identical, r = 1, and
Cvtota z =
~ pCv2 1 + (1-p)Cv22
so the total is the root mean square of the weighted individual component variabilities.
Thus, Cvtota1 is guaranteed only to be the minimum of Cv1and Cv2 . In most geological
situations, r =F 1 because the elements have differences in the energies, source materials,
and other factors that produced them. When r >> 1 andp < 1,
Statistics for Petroleum Engineers and Geoscientists
149
so the variability of the larger mean population dominates and is amplified by
(1-p)-112. For these cases, Cy 1 may be near Cy + Cy2 •
1
tota
One case where Cy's are additive concerns the sensitivity analysis of estimators to
errors in the data (Chap. 5). We saw that random variables that are products of other
random variables give' an especially simple form to the sensitivity equation. For
example, if m and n in Archie's law (Chap. 5) are constant, the sensitivity of Sw to errors
in Rw, Rt, a, and qy is
If we interpret the L1's as being the standard deviations of these variables, we can recast the
above as
Each term has the form of a variability over a true value, giving it the same form as a
coefficient of variation. Hence, we can express the variability of Sw in the following
manner:
1
Cvs w =-n (CvR w
+
CvR t +
Cy a + m
Cv,.)
'Y
Consequently, we can see at a glance which measurement(s) will contribute the most to
the uncertainty in Sw. A similar treatment can be made for other equations, such as the
STOIIP expression discussed in Chap. 3.
The above expression for the total C v is an approximation that works best for small
Cy's (i.e., Cy < 0.5). It will show the relative contributions of each component of
variability. A better approximation applies if the variables are all log-normally
distributed in a product. Recall from Chap. 4 that if ln(X) - N(J.lx, q 2), then O"x 2 =
ln(l +
C:V~ ).
Hence, if ln(Y)- N(J.lx,
~ 2) and Z = XY,
then
Chap. 6
150
2
O"z =
Solving for
2
O"x
+
2
O"Y
2
.
Measures of Heterogeneity
2
2
and ln(l + Cv) = ln(l + Cv) + ln(l + Cyy)
c;.,2z gives
2
2
2
2
2
22
Cy = (1 + Cy )(1 +Cy)- 1 = Cy + Cy + Cy Cy
%
y
%
y
%
y
Neglecting cross terms and extending to products of I terms, we obtain the
approximation
We also note that, because Cy = [exp(o-2)- 1]112 for ln(k)- N(j.l, o-2), Cy is an
estimator for the standard deviation o-when o-is small (e.g., O"< 0.5).
Cy can also be used to guide sampling density. The so-called "N-zero method,"
discussed by Hurst and Rosvoll (1990), is based on two results of statistical theory:
1.
The Central Limit Theorem states that, if Is independent samples are drawn from
a population (not necessarily normal) with mean Jl and standard deviation 0", then
the distribution of their arithmetic average will be approximately normal.
2.
The sample average will have mean J.L and standard error
ot-{4.
See Chap. 5 for more details. From these two points, the probability that the sample
average ( k 8 ) of Is observations lies within a certain range of the population mean (j.l) can
be determined for a given confidence interval.
For a 95% confidence level, the range of the average is given by± t • SE, where the
J';-{1;.
standard error (SE) is approximated by
The larger the sample number I 8 , the more
confident we can be about estimates of the mean. SE is the standard deviation of the
sample mean, drawn from a parent population, and is a measure of the difference between
sample and population means.
The student's tis a measure of the difference between the estimated mean, for a single
sample, and the population mean, normalized by the SE. For normal distributions, the t
Statistics for Petroleum Engineers and Geoscientists
151
value varies with size of sample and confidence level (Chap. 5). The above statement can
be expressed mathematically as
Consider now another sample of size / 0 such that k 0
predetermined confidence interval:
Pk 0 )
Prob [(J1 - 100 ~ k o ~
(
± P%
tolerance satisfies the
Pk 0 )~U = 95%
J1 + 100
This time, we have expressed the permissible error in terms of a percentage of k 0 • When
both conditions are satisfied, Is =I0 and
Pko
s"
-=t-
100
Wo
Rearranging this gives an expression for the appropriate number of specimens, / 0 :
I
o
=(lOOt; )2
(6-1)
Pko
where the nearest integer value for the right side is taken for 10 • For Is> 30, t =2 and
with a 20% tolerance (i.e., the sample mean will be within ±20% of the parent mean for
95% of all possible samples, which we consider to be an acceptable limitation), this
expression reduces to ·
" s/k
"_
(2oocv )2 where Cv=
lo = . 20
0
or
This rule of thumb is a simple way of determining sample sufficiency. Of course, since
" is a random variable, I 0 is .also random in the abov~ expression. I 0 will change
Cv
because of sampling variability. The above is called the I 0 -sampling approach.
152
Chap. 6
Measures of Heterogeneity
Although derived for the estimate of the arithmetic mean from uncorrelated samples by
normal theory, we have found it useful in designing sample programs in a range of core
and outcrop studies. Having determined the appropriate number of samples, the domain
length (d) will determine the sample spacing (d0 ) as d0 = d I 10 •
An initial sample of 25 measurements, evenly spaced over the domain, which can be a
" estimated from this
lamina, bedform, formation, outcrop, etc., is recommended. If Cy,
sample, is less than 0.5, sufficient samples have been collected. If more are required,
infilling the original with 1, 2, or J samples will give 50, 75, or 25! samples. In this
way, sufficient samples can be collected.
Example 2 - Variability and Sample Size. In a 9-ft interval of wavy bedded
material (Fig. 6-3), the performance of core plugs is compared with the probe
permeameter for the estimate of the mean permeability of the interval. Based
"
.
on the core plugs, Cy= 0.74, giving 10 =55 and d0 = 2 in., well below the
customary 1-ft sample spacing. The probe data give/0 = 98 and d0 = 1 in.
For such variability, about 100 probe-permeameter measurements are needed
for estimates within ±20% but, because of the way core plugs are taken
(Chap. 1), plugs are an impractical method for adequately sampling this
interval. Nine plugs, taken one per foot over the 9-ft interval, are clearly
insufficient even if they are not biased towards the high-permeability
intervals.
Comparing the sample means, the plug estimate is 2.3 times the probe value.
Why are they so different? Equation (6-1) can be rearranged to solve for P for
" .
both data sets. In the case of the nine plug data, Cv= 0.74 and t = 2.3, so
that P = 57%. The true mean is within 390 roD ± 57% about 95% of the
time. A similar calculation for the probe data shows the mean to be in the
range 172 mD ± 12%. Thus, the estimates are not statistically different.
That is, the difference between the estimated means can be explained by
sampling variability. No preferential sampling by the plugs in the better,
higher-permeability material is indicated. The plugs have simply
undersampled this highly variable interval.
Corbett and Jensen (1992a) have suggested that Cy's in clastic sediments could have
sufficient predictability that sample numbers could be estimated by using the geological
description. While further data are needed, there are indications that stratal elements of
similar depositional environments have similar variabilities. For example, the two
fluvial trough crossbed data sets of Fig. 6-2, obtained at different locations of the same
outcrop, show very similar Cy's. Thus, applying the / 0 -sampling approach, sample
Statistics for Petroleum Engineers and Geoscientists
153
numbers could be predicted on the basis of the geological element (e.g., aeolian dune
foreset, fluvial trough crossbed, fluvio-marine tidal channel).
0
+
Core plug
Probe permeameter
PLUGS: I= 9 data
1\
Cv = 0.74, Arith. av. = 390mD
PROBE: I= 274 data
1\
Cv = 0.99, Arith. av. = 172mD
-xx58
10
Figure 6-3.
100
1000
Permeability, mD
Core-plug and probe-permeameter data in a Rannach formation interval.
Dykstra~Parsons
Coefficient
The most common measure of permeability variation used in the petroleum industry is
VDP• the Dykstra-Parsons coefficient (Dykstra and Parsons, 1950):
v
_ ko.so- ko.16
DP-
ko.so
(6-2)
where ko.so is the median permeability and k0 . 16 is the permeability one standard
deviation below ko.so on a log-probability plot (Fig. 6-4). Vvp is zero for homogeneous
reservoirs and one for the hypothetical "infinitely" heterogeneous reservoir. The latter is a
layered reservoir having one layer of infinite permeability and nonzero thickness. VDP
has also been called the coefficient of permeability variation, the variance, or the
variation. Other definitions of VDP involving permeability-porosity ratios and/or variable
sample sizes are possible (Lake, 1989, p. 196).
Chap. 6
154
Measures of Heterogeneity
The Dykstra-Parsons coefficient is computed from a set ofpermeability data ordered in
increasing value. The probability associated with each data point is the thickness of the
interval represented by that data point. Dykstra and Parsons (1950) state that the values
to be used in the defmition are taken from a "best fit" line through the data when they are
plotted on a log-probability. plot (Fig. 6-4). Some later published work, however, does
not mention the line and depends directly on the data to estimate ko.l6 and ko.so (Jensen
and Currie, 1990).
1\
1\
ko.so - ko.16
A
ko.so
·-:s
~
....
~
-
~
~~--------------~
0.50
~
~
0.02
0.16
0.50
Probability Scale
Figure 6-4.
0.84
0.98
p
Dykstra-Parsons plot.
There are several drawbacks in using the "best fit" approach, especially if the data set is
not log-normally distributed. Jensen and Lake (1988) demonstrate the nonunique
properties of VDP in light of an entire family of p-normal distributions. Although
Dykstra and Parsons were able to correlate VDP with expected waterflood performance,
Statistics for Petroleum Engineers and Geoscientists
155
caution should be exercised in the universal application of VDP for data sets that are not
log-normal. See, for example, Willhite's (1986) Example 5.6.
From the definition for VDP (Eq. (6-2)), VDP = 1 - e-cr when ln(kl;- N(Jl, a 2). Thls
relation"between VDP and a provides an alternative VD)l. estimator: V DP = 1 - exp(-s),
"Xhere s is the sample standard deviation of ln(k). V DP is about twice as efficient as
VDP (Jensen and Currie, 1990).
A study by Lambert (1981) shows that VDP estimated from vertical wells ranges
between 0.65 and 0.99. VDp's measured areally (from arithmetically averaged well
permeabilities) range from 0.12 to 0.93. Both the areal and vertical VDP's are normally
distributed, even though most of the permeability values themselves are not normally
distributed. Of greater significance, however, is the observation that VDP did not
apparently differentiate between formation type. This observation suggests that either
VDP is, by itself, insufficient to characterize the spatial distribution of permeability
and/or the estimator itself is suspect. Probably both suggestions are correct; we elaborate
on the latter theme in the next few paragraphs.
The simplicity of VDP means that analytical formulas may be derived for the bias and
standard error (Jensen and Lake, 1988), assuming ln(k) is normally distributed. The bias
is given by
bv=- 0.749[ln(l- VDp)] 2 (1- VDp) I I
and the standard error is
sv= -1.49[ln(l- VDp)](l- VDp)/fi
1\
where I is the number of data in the sample. The bias is always negative (VDP
underestimates the heterogeneity), is inversely proportional to I, and reaches a maximum
in absolute value when VDP = 0.87. However, the bias is generally small. For example,
when VDP = 0.87, only I= 40 data are required to obtain by=- 0.009.
1\
The variability of VDP• on the other hand, can be significant (Fig. 6-5). The standard
error decreases as the inverse of the square root of I and sy attains a maximum at
VDP = 0.63. The number of data needed to keep sv small is quite large for moderately
heterogeneous formations. For example, at VDP = 0.6, 120 samples are required to attain
a precision of sv = 0.05. From the definition of standard error, this value means that
Chap. 6
156
Measures of Heterogeneity
Vnp has a 68% (±one standard deviation) chance of being between 0.55 and 0.65,
ignoring the small bias.
-s
1000
<l)
~
Sy=0.02
~
"-!
....=
.... 100
't:l
~
~
0.05
s
c;..
0
~
,Q
z=
10
0.2
0.4
0.6
0.8
1
Vnp
Figure 6-5.
Variability performance of Vnp estimator. From Jensen and Lake (1988).
Example 3- Estimating Vnpfor a Data Set. Estimate Vnp for the data set
in Example 4, Chap. 3: 900, 591, 381, 450, 430, 1212, 730, 565, 407,
440, 283, 650, 315, 500,420, 714, 324. Assume each datum represents the
*
same volume of the reservoir. The ordered data set (xj) and its logarithm (xi)
are shown in the table below, repeated from Example 4.
Figure 6-6 shows the probability plot for x *. From the "best fit" line
A
established by eye, VDP = (493 - 308) I 493 = 0.38. These data represent a
A
fairly homogeneous reservoir. If we assume that VDP = VDP• then bv = -0.006
and sv = 0.11. While bv is negligible, sy is sufficiently large that rounding
A
VDP to 0.4 is appropriate.
As we have stated before, statistics such as VDP can gloss over a number of
details. Even in this example, with very few data, this is true. The CDF
presented above manifests evidence of two populations; a small-value sample
that is even more homogeneous than VDP = 0.4, and a large-value portion
that is about as homogeneous as the entire sample.
Statistics for Petroleum Engineers and Geoscientists
No.
1
2
3
4
5
6
7
8
9
*
X;
xi
283
315
324
381
407
420
430
440
450
5.65
5.75
5.78
5.94
6.01
6.04
6.06
6.09
6.11
157
No.
10
Prob., Pi
0.029
0.088
0.147
0.206
0.265
0.324
0.382
0.441
0.500
11
12
13
14
15
16
17
*
X;
Xi
500
565
591
650
714
730
900
1212
6.21
6.34
6.38
6.48
6.57
6.59
6.80
7.10
Prob., Pi
0.559
0.618
0.676
0.735
0.794
0.853
o.9T~
0.971
7.0
M
6.5
493
1:1>.0
0
1\
= k 0_50
~
6.0
"
5.5
-3
-2
-1
1
0
2
3
97.7
99.9
Normal Quantiles
0.1
2.3
16
84
50
Probability (%)
Figure 6-6.
A
Probability plot for VDP in Example 3.
Chap. 6
158
Measures of Heterogeneity
A number of investigators have used VDP to correlate oil recovery from core-scale
waterfloods (see Lake and Jensen, 1991, for examples). A significant feature of many
studies is the small sensitivity of models to variations in VDP when VDP :::;; 0.5 while,
for the large-heterogeneity cases, there is a large sensitivity. This behavior reflects, in
part, that V DP• with a finite range of 0 to 1, poorly discerns large-heterogeneity
situations.
lorenz Coefficient
A less well-known but more general static measure of variability is the Lorenz
coefficient, Lc. To compute this coefficient, first arrange the permeability values in
decreasing order of k/1/J and then calculate the partial sums.
where 1 :::;; J ~I and there are I data. We then plot F versus Con a linear graph (Fig. 6-7)
and connect the points to form the Lorenz curve BCD. The curve must pass through (0,
0) and (1, 1). If A is the area between the curve and the diagonal (shaded region in Fig. 67), the Lorenz coefficient is defined as Lc =2A. Using the trapezoidal integration rule,
we have (Lake and Jensen, 1991)
ic =
I J k
1
(J-1)2
k·
I. I. ..J.. _ _j_
f~ i=lj=l
c/Ji
c/Jj
i=li/Ji
Just as for VDP• Lc is 0 for homogeneous reservoirs and 1 for infinitely heterogeneous
reservoirs, and field-measured values of Lc appear to range from 0.6 to 0.9. However, in
general, VDP i' Lc.
The Lorenz coefficient has several advantages over the Dykstra-Parsons coefficient.
Statistics for Petroleum Engineers and Geoscientists
159
Fraction of Total Storage Capacity
Figure 6-7.
Lorenz plot for heterogeneity.
1.
It can be calculated with good accuracy for any distribution. However, for the
family of p-normal distributions, Lc is still not a unique measure of variability.
2.
It does not rely on best-fit procedures. In fact, being essentially a numerical
integration, there is typically less calculation error in Lc than in VDP·
3.
Its evaluation includes porosity heterogeneity and (explicitly) variable thickness
layers.
Lc is, however, somewhat harder to calculate than VDP and has not, as yet, been
directly related to oil recovery. For a reservoir consisting of I uniformly stratified
elements between wells through which is flowing a single-phase fluid, the F-C curve has
a physical interpretation, one of the few such instances of a physical meaning for a CDF.
F represents the fraction of the total flow passing a fraction C of the reservoir volume.
For example, in Fig. 6-7, approximately 80% (F = 0.8) of the flow is passing through
50% (C = 0.5) of the reservoir. This curve, therefore, plays the same role, at a large
scale, that a water-oil fractional flow curve does on a small scale, which accounts for its
use in the same fashion as the Buckley-Leverett theory (Lake, 1989, Chap. 5).
Chap. 6
160
Measures of Heterogeneity
For ln(k) -N(/1., ci2), Lc. Vvp. Cv, and a are related by the following expressions (if
porosity is constant):
where erf() is the error function discussed in Chap. 4. Lc is also known as Gini's
coefficient of concentration in statistical texts (Kendall and Stuart, 1977, p. 48).
A
A
It is more difficult to develop estimates of bias and precision for Lc than for V DP·
Using numerical methods, however, we can still present the results graphically if we
assume ln(k) is normally distributed. Figure 6-8 presents the bias as a fraction of the true
value in Lc. The x axis gives the number of data in the sample, and each curve is for
different values of Lc beginning at 0.3 (topmost curve) down to 0.9 (lowest curve). The
bias can be pronounced, particularly at high Lc and small sample sizes. For example, for
I= 40 data in the sample and a true Lc =0.80, repeated measurements will actually yield
A
an estimated Lc of 0.72 (since the bias is -0.08). Lc is, on average, lower than the true
value; thus, we are again underestimating the heterogeneity.
0.0
-0.1
I.e =0.3
-0.2
I.e= 0.6
0
0
25
50
Lc=0.9
75
100
Number of data in sample
Figure 6-8.
Bias performance of the Lorenz coefficient estimator. Lines are best-fit
curves for Lc = 0.3 and 0.9. From Jensen and Lake (1988).
Statistics for Petroleum Engineers and Geoscientists
i 6i
/\
Figure 6-9 gives the error sL (standard error in Lc) for Lc. For example, when
Lc = 0.8 (neglecting the correction for bias), about 140 data values are required to
determine Lc to within a standard error of 0.05. Do not try to compare this figure with
the similar figure for VDP• because sL is a measure of the error in the Lorenz estimate in
units of Lc and sy measures error in units of VDP· Since VDP Lc;., in general, the
units are not the sa~e. For an equivalent situation, it turns out that Lc usually has a
lower error than the VDP estimate. See Jensen and Lake (1988) for further details.
'*
-e
Q)
!:l.
1000
~
....=
.....e1:ie1:i
~
100
~
0
r..
~
.Q
5
z=
10
0.2
0.4
0.6
0.8
1.0
Lc
Figure 6-9.
Variability of Lorenz coefficient estimates. After Jensen and Lake (1988).
Gelhar~Axness
Coefficient
A combined static measure of heterogeneity and spatial correlation is the Gelhar-Axness
coefficient (Gelhar and Axness, 1979), defined as
Chap. 6
162
where
Measures of Heterogeneity
a~(k) = Var[ln(k)] and An is the autocorrelation length in the main direction along
which flow is taking place, expressed as a fraction of the inlet to outlet spacing. An is a
measure of the distance over which similar permeability values exist. We discuss An in
more detail in Chap. 11.
IH has the great advantage of combining both heterogeneity and structure in a quite
compact form. Cv. VnP• and Lc all ignore how the permeability varies in the rock;
regions of high and low permeability could be far apart or quite close and yet have the
same Cv, VnP• and Lc. As we have discussed, however, Cv with the geological
description does appear to have potential for guiding sampling strategies. I H begins to
incorporate a numerical measure of the geological organization present in the rock. As
with VnP• IH asumes ln(k) is normally distributed.
Numerical simulation work by Waggoner et al. (1992) and Kempers (1990) indicates
that!His a much improved indicator of flow performance compared to VnP· However, it
cannot be satisfactory for all cases since, in the limit of An ~oo (uniform, continuous
layers), the flow must again be governed by VnP· Sorbie et al. (1994) discuss some
limitations of I H• particularly as it relates to prediction of flow regimes for small An.
6=3
DYNAMIC
MEASURES
In principle, these measures should be superior to static measures since they most directly
characterize flow. In practice, dynamic measures are difficult to infer and, like the static
measures, are unclear as to their translation to other systems and scales.
The measures we discuss are based on end-member displacements. A displacement that
has an uneven front is called channeling and its progress is to be characterized by a Koval
factor. If it has an even front it is dispersive and is characterized by a dispersion
coefficient. See Chap. 13 and Waggoner et al. (1992) for more details.
Koval Factor
The Koval heterogeneity factor H K is used in miscible flooding to empirically incorporate
the effect of heterogeneity on viscous fingering (Koval, 1963). It is defined as the
reciprocal of the dimensionless breakthrough time in a unit-mobility-ratio displacement:
HK = l/tn breakthrough
Statistics for Petroleum Engineers and Geoscientists
163
It follows from this that the maximum oil recovery has been obtained when H K pore·
volumes of fluid have been injected:
tV sweepout -H
K
In these equations, tv is the volume of displacing fluid injected divided by the pore
volume of the medium. Breakthrough means when the displacing fluid first arrives at the
outlet and sweepout means that the originally resident fluid is no longer being produced.
For a homogeneous medium, both occur at tv= 1 andHK= 1, but there is no upper limit
on HK. Obviously, large values of HK are detrimental to recovery.
If we interpret the above response as occurring in a uniformly layered medium with a
log-normal permeability distribution, then H K and VVP are empirically related by
which plots as Fig. 6-10.
25
log10 (HI()
20
= V 0 p(1-V0 pr0·2
Calculated
15
10
5
o~----~----~----~----~~--~
0
0.2
0.4
0.6
0.8
1
Vnp
Figure 6-10.
Koval factor and Dykstra-Parsons coefficient for a uniformly layered
medium. From Paul et al. (1982).
Chap. 6
164
Measures of Heterogeneity
H K has many of the same problems as VDP and Lc. but it is a far more linear measure
of performance, being bounded between zero and infinity. If the medium is not uniformly
layered, HK can be related to IH , the Gelhar-Axness coefficient, as in Fig. 6-11.
10
Koval Factor, HK
1
0.0001
0.001
0.01
0.1
1
10
Heterogeneity Index, IH
Figure 6-11.
Relationship between Koval factor and Gelhar-Axness coefficient From
Datta Gupta et al. (1992).
Dispersion
If a miscible displacement proceeds through a one-dimensional "homogeneous" permeable
medium, its concentration at any position x and time tis given by
C(x, t)
1[
=2
1 - erf
(x-vt)~
~ U
2
where v is the mean interstitial velocity of the displacement. The most important
parameter in this expression is Kz, the longitudinal dispersion coefficient. Kz has been
found to be proportional to v according to
Statistics for Petroleum Engineers and Geoscientists
165
where a 1 is the longitudinal dispersivity, an intrinsic property of the medium. Clearly,
the degree of spreading depends on the dispersivity; the larger a 1, the more spreading
occurs.
The derivation and implications of the equation for C(x,t) are too numerous to discuss
here (Lake, 1989, pp. 157-168). However, both HK and a 1 are clearly manifestations of
heterogeneity in the medium because there would be no spreading of the displacement
front if the medium were truly homogeneous. Indeed, several attempts have been made
to relate a 1 to the statistical properties of the medium. In the limiting case of small
autocorrelation and heterogeneity, a1 divided by the length of the medium is proportional
to I H (Arya et al., 1988). This should come as no surprise if we view such dispersion as
a series of uncorrelated particle jumps during the displacement. In such circumstances,
the average particle position, now being the result of a series of such jumps, should take
on a Gaussian character as required by the Central Limit Theorem (Chap. 4) and suggested
by the error function in the C(x,t) equation.
Unfortunately, a substantial amount of data collected in the field suggests that a 1
depends on the scale of the measurement. Such behavior is called non-Fickian dispersion.
In such cases, a 1 can still be related to the statistical properties of the medium (Dagan,
1989, Chap. 4; Gelhar, 1993), but the connection is much more involved and its utility
in subsequent applications is reduced.
Datta Gupta et al. (1992) have shown that the concentration profile of miscible
displacements through a correlated medium behaves as a truncated Gaussian distribution,
becoming more Gaussian as the scale of correlation decreases with respect to the length of
the medium. Adopting this point of view, the dispersion relation (with constant az) and
Koval approach (with constant H K) represent the extremes of small and large spatial
correlation, respectively. (If we applied the Koval approach to uncorrelated media, we
would find that H K depends on scale.) Nevertheless, both approaches give material
properties (a 1 or H K) that are related to the heterogeneity of the medium.
6-4
SUMMARY
REMARKS
The progression of this chapter mimics the progress of the entire book in a way. We
have begun by discussing heterogeneity measures and their properties. But during the
dynamic measure discussion, we could not avoid speaking about autocorrelation or spatial
arrangement. This is a subject we will return to later in the book. But we must depart
this chapter with a bit of foreshadowing.
166
Chap. 6
Measures of Heterogeneity
The subject of autocorrelation (and its existence in permeable media properties) means
that measures of heterogeneity must be scale-dependent. Indeed, laborious measurements
in the field have shown this to be the case (Goggin et al., 1988). There are entire
statistical tools devoted to such measurements, and these tools frequently combine spatial
arrangement and heterogeneity. Chapter 11 contains a discussion of these, along with the
implications for the geological character.
Spatial arrangement implies some force or causality behind the observation. Since we
are dealing with naturally occurring media, a study and understanding of this causality
must deal with the geologic reasons behind the observations. Indeed, we shall fmd that
geologic insight can frequently unravel a statistical conundrum, allow us to fill in
missing data, or even indicate that a statistical approach is not necessary. Such a
combination is most powerful and forms an underlying theme for this book.
7
HYPOTHESIS TESTS
Every computation produces an estimate of some formation property. Given unlimite~
resources and ·time, we could very accurately establish the exact value of that property
using the entire volume of material we wish to assess. Since resources and time are
limited, the values we calculate use modest sample numbers and sample volumes and may
differ from the exact value because of sampling variability. That is, if we could obtain
several sets of measurements, the estimates would vary from set to set because each
estimate is a function of the data in that set. An estimate will have an associated error
range, the standard error, which gives an idea of the precision of the estimate attributable
to sampling fluctuation. The standard error is a number based on the number and
variability of the measurements and does not take into account biased sampling procedures
or other inadequacies in the sampling program. Chapter 5 described some methods for
determiningstandarderro~.
In this chapter, the central theme is making comparisons involving one or more
random variables. A hypothesis test (or a confidence test) is a formal procedure to judge
whether some estimate is different in a statistical sense from some other quantity. That
latter quantity can be a second estimate or some number ("truth") obtained by another
method. We will use several methods for comparing estimates. Whatever the technique
used, the comparison is made to answer the question: can the difference in value between
the two quantities be explained by sampling variability? If the answer is yes, then we say
the two quantities are not statistically different.
In many cases, hypothesis tests may be unnecessary because they do not answer the
appropriate question or they only formalize what we already knew. For example, suppose
we have two laterally extensive facies,f1 andf2 , with/1 abovef2 • For the purposes of
167
Chap. 7
168
Hypothesis Tests
developing a flow-simulation model, we want to know whether they can be combined or
if they should be separately distinguished; Assuming that the arithmetic average is the
appropriate average for flow in each of these facies, we calculate the average
permeabilities and standard errors from core samples and obtain the results
k; =
faciesf1:
faciesj2:
30
± 10 mD
kz = 100 ± 20 mD
Do we need a hypothesis test to establish that/1 and/2 are statistically different? No! A
simple sketch shows us that the 95% confidence intervals do not intersect (Fig. 7-1).
• • •
95%
\\.
\i:
0
Figure 7-1.
10
50
I
I
kl =30
95%
•
140
60
I
I
.k
k2= 100
Two arithmetic averages and their 95% confidence intervals (twice their
standard errors).
So the statistical significance of the averages is . clear: h and h have average
permeabilities that are different at the 95% level. Wha~ is less evident is whether, under
the desired flow process, separating the facies into two strata will improve the predictions
made by the simulator model. This issue is beyond the ability of hypothesis testing to
answer.
Hypothesis tests are also unsuitable for analyzing noncomparable quantities. One
indicator of comparability is whether the two quantities have the same units (dimensions),
but this test is incomplete. The quantities should be additive (see Chaps. 3 and 5)
because hypothesis tests assume that differences in estimates are meaningful. For
example, consider the two regions shown in Fig. 7-2. If they have equal areas of the
front and back faces.for injection or production and no-flow boundaries otherwise, they
have identical abilities to transmit a single phase. Thus, while a hypothesis test may
indicate that the permeabilities of the two regions are statistically different, their
transmissibilities are equal. This problem arises because permeability is ah intensive
property and, therefore, is not additive. Flow resistance or conductance, however, may be
additive (Muskat, 1938, pp. 403-404) so that, with similar boundary conditions, the
sample statistics of these variables may be comparable. It is easy to overlook this aspect
of hypothesis testing in the heat of the statistical battle.
Statistics for Petroleum Engineers and Geoscientists
169
No-flow surfaces
Figure 7-2.
Two samples with differing average permeabilities but similar flow
properties.
Statisticians have implicitly recognized the need for additivity in hypothesis testing.
Their 'concern, however, usually focuses on knowing the PDF of the difference of the
quantities to be compared. For example, problems are often expressed so that two
normally distributed variables are compared because the difference of two uncorrelated
normal variables is also normal. Besides the normal, the c2 and the Poisson distributions
also have this property (see Cramer, 1946, and Johnson and Kotz, 1970, for more), but
most other PDF's do not. Because the c2 PDF is additive and variances of independent
variables are additive, hypothesis tests on variabilities compare sample variances rather
than standard deviations. Here, we will take the approach discussed in Chap. 5 and require
only local additivity (i.e., small perturbations are additive even though large perturbations
may not be additive).
Hypothesis tests are similar in some ways to the judicial process that exists in many
countries. When someone is accused of committing a crime, a trial is held to determine
the guilt or innocence of the accused. The accused is assumed to be innocent until a jury
decides. At the conclusion of the trial, there are usually four possible results to this
process:
1.
2.
3.
4.
the accused is actually guilty and is found guilty;
the accused is actually innocent and is found innocent;
the accused is actually innocent but is found guilty; and
the accused is actually guilty but is found innocent.
In the first result, the assumption (innocence) was wrong and was found to be wrong. In
the second result, the assumption was right and was confirmed. In both cases, the
outcomes reflected the true status of the accused and justice was served. In the last two
Chap. 7
170
Hypothesis Tests
possibilities, however, mistakes were made and the outcomes were wrong. Different
countries handle these errors differently. The judicial process may be adjusted to
minimize outcome 3 at the expense of increasing the occurrences of outcome 4 by, for
example, increasing the burden of proof required for a conviction.
For hypothesis tests, we start with an assumption that is called the null hypothesis.
Sampling variability alone often can explain the difference betweenJ:wo em:imates. For
example, faciesfi andh have the same mean permeability and k 1 =F- k 2 because of
sampling variation. In this case, the alternative hypothesis is that the difference cannot
be explained by sampling variability and some other cause (unspecified) must be in effect.
We then apply the hypothesis test and obtain a result that either upholds the null
hypothesis or shows that the alternative is indicated. Just as in the judicial process,
mistakes may result. We may decide the difference in two quantities cannot be explained
by sampling variability when, if the truth could be known, it could be so explained
(judicial outcome 3). This type of mistaken conclusion is called a type-I error. A type-2
error occurs when the null hypothesis is accepted when, in fact, the altema,tive applies
(judicial outcome 4). Before we apply a hypothesis test, we usually stipulate the
probability of avoiding a type-1 error, and this probability is called the confidence level of
the test.
·
Depending on the hypothesis test used, the number of errors will vary. Different tests
may give different outcomes. The choice of test depends upon the information available,
the estimates to be tested, and the consequences of making errors. We will only consider
a few hypothesis tests and their properties that are commonly used to analyze
geoscientific and geoengineering data. We will assume that, when comparisons are made
between different sample statistics, the statistics are based on independent samples.
Extensive treatments are given by Rohatgi (1984), Snedecor and Cochrane (1980), and
Rock (1988). Cox and Hinkley (1974) discuss the theoretical underpinnings of
hypothesis tests.
7-1
PARAMETRIC HYPOTHESIS TESTS
These tests depend on a knowledge of the form of the PDF from which the data come.
Common tests involve the normal, binomial, Poisson, and uniform distributions. The t
and F tests are particularly common and assume the data come from normal PDF's.
The
t Test
Populations A and B will be assumed normal with means and variances (J.LA• cr~) and
(J.l.B• cri). We begin with the case cr~ = 0'~ =
and deal later with the case cr~ =F- cri,
i
Statistics for Petroleum Engineers and Geoscientists
171
From the data sets having IA and IB data, respectively, we compute ( XA.~~) and
( XB, ~i) and make the null hypothesis H0 that the two population means are equal,
null hypothesis, H o= llA = llB
with the alternative hypothesis HA that the two means are not equal,
alternative hypothesis, H A: llA i'JlB
If we can demonstrate statistically that, at some predetermined confidence level Ia, the
null hypothesis is false, then we have found a "significant difference" in the sample
means ( XA - XB). That is, it is a difference that, at the given level of probability,
cannot be explained by sampling variability alone. The roles of the hypotheses and
confidence level are quite important because they govern what question the test will
answer and with what probability type- I errors will occur. ( XA - XB ) has a t
distribution with (IA + IB - 2) degrees of freedom and a variance of 0' 2
+ I"B\ Since
H A is satisfied if either llA > llB or llA < llB, we have to consider both possibilities
when choosing the desired confidence level. That is, there is a probability of a/2 that llA
> llB and a probability of a/2 that llA < llB· Such a situation requires a "two-tailed" t
test.
(r}
The appropriate statistic is
where
We reject Ho if
Prob[ltl > t(a/2,IA +IB- 2)] >a
Chap. 7
172
Hypothesis Tests
and we acceptH0 if
Prob[ ltl > t(a/2,/A +IB- 2)]::;;; a
In terms of confidence intervals, H 0 is accepted if
where to= lt(a/2, /A+ IB- 2)1. If the alternative HA had admitted only one inequality
(e.g., JlA > JlB), then the critical t value would be to= t(a, !A+ IB- 2).
Example la- Comparing Two Averages (Equal Variances Assumed).
Consider the following data:
XA =50
For a= 0.05 (95% confidence level), t(a/2,
values (Table 5-2).
df> =2.006 from a table oft
~2 = 24•5 + 29•3 = 3.91
25 + 30 - 2
t=
50- 48
--;:::::::=====:....=
~ 3·91 (is + io)
3.74
Since the absolute value oft is greater than 2.006, we reject H 0 in favor of
HA and conclude with 95% confidence that the means are not equal. Recall
from Chap. 5, J1 == X± t(a/2, df)s/{/, and this situation can be represented
as in Fig. 7-3.
Unfortunately, errors are the inevitable byproduct of hypothesis testing. No matter
how careful we are in selecting the form of the null hypothesis and associated confidence
level, we always run the risk of rejecting Ho when it is true or accepting Ho when HA is
true. Once an inference is made, there is no way of knowing whether an error was
Statistics for Petroleum Engineers and Geoscientists
173
committed. However, it is possible to compute the probability of committing such an
error using the t distribution. In fact, since probabilities and frequencies are equivalent,
we can say that if we repeat the above test many times, we will find the sample means to
be different 95 times out of 100 trials.
...
95%
0
~
XB 48.65
'\ 47.35
'" I
~
I
I
95%
XA
49.08
I
I
50.92
I
50.0
48.0
t(0.025, 24) = 2.064
t(0.025, 29) = 2.045
Figure 7-3.
.
Relationships of averages and their 95% confidence limits. Note that
= 0.92.
2.045 {3 I f30 = 0.65 and 2.064 {5 I
m
The above analysis is largely unchanged when
combine the variance estimates
and s~to give
freedom. The appropriate statistic is
sx
~
=F
o-~ except that we can no longer
s and we have to adjust the degrees of
t=
We reject Ho if Prob[ ltl > t(a/2, d.[)]> a and accept Ho if Prob[ ltl > t(a/2, d.[)]:; a
with
as an "adjusted" degrees of freedom that has been devised to allow us to continue using the
conventional t tables (Satterthwaite, 1946).
Chap. 7
174
Hypothesis Tests
Example 1b - Comparing Two Averages (Unequal Variances Assumed).
Suppose that we have the same data as before but the populations are no
longer presumed to have the same variance:
0.090
0.0020 = 44.7
For a= 0.05 (95% confidence level), t(a/2, df>
t = 50 - 48
~is+ :o
=
44
= 2.015 (from Table 5-2) and
= 3.65
Since the absolute value oft is greater than 2.015, we reject H 0 in favor of
HA and conclude with 95% confidence that the means are not equal.
In the t test, additional information has direct value in terms of the number of data
required. For example, in Example lb, the number of degrees of freedom decreases from
53 to 44 when we remove the information that cr~ =
increasing slightly the
possibility of a type-1 error (i.e., a computed ltl that is closer to t(a/2, df)). This is in
keepinf with the intuitive notion that information has value. In this case, the knowledge
that O'A = 0'~ is worth 9 data points. Similarly, if we knew that JlA > Jln was
impossible from some engineering argument (e.g., a law of physics, thermodynamics, or
geology), we could use a one-tailed t test instead of a two-tailed test In Example la, the
critical t value, t(O.OS, 53), would then be 1.67 instead of 2.006. Alternatively, to have a
critical t-value of 2.006, the df could be reduced to 6, so we would need only 8 data (df =
I A + I B - 2). The information that JlA > f.ln can be excluded is therefore worth 45 data
points in this case.
0'1,
The t-test theory is developed on the assumption of normality. It has been found,
however, that the test is rather insensitive to deviations from normality (Miller, 1986).
For the arithmetic average, the Central Limit Theorem (Chap. 4) suggests that
Statistics for Petroleum Engineers and Geoscientists
XA
175
(= .!. f Xi) will be much closer to normal than the population from which the
I i=l
samples Xi are taken. This argument, however, may not apply when other quantities,
such as the geometric ( Xc) or harmonic ( XH) averages, are being compared. In that
case, transformat~ns might be useful. For example, if the_!! i are log-normally
distributed, ln( Xc) has a normal PDF, so comparing In( Xc)'s is appropriate.
Similarly,
xfi is likely to be more nearly normal than
XH because of the central limit
effect, so the inverse of harmonic averages may be compared with the t test. If
transformed quantities are being compared, however, the standard deviations also have to
pertain to the transformed quantity. Box et al. (1978, pp. 122-123) give an example of
applying at test to a log-transformed data set. Transformation before the t test, however,
will not work if one average is being compared with another type of average, e.g., XA
from one facies (approximately normal) compared with XH of another facies (nonnormal) for some property X. If severe nonnormality of the two quantities is suspected
and they really are comparable, a nonparametric method, such as the Mann-WhitneyWilcoxon test (see below) may be used.
When three or more quantities are to be compared, it may be tempting to apply the t
test pairwise to the estimates. In general, this is not advisable because of the increasing
chance of a type-1 error. If we compare two estimates at the 95% confidence level, there
is a 1 - 0.95 = 0.05 probability that H 0 is. satisfied but HA is indicated. If we compare
pairwise three estimates at the 95% level, there is a 1 - (0.95)3 = 0.14 probability or a
14% chance that H0 is satisfied but HA will be indicated by at least one pair of estimates.
The preferred approach to testing several estimates is to use the F test. However, the
elementary application of the F test is to compare population variances, a subject we deal
with next.
The F Test
We begin again with e1e now familiar assumption that populations A and B are normally
distributed with means and variances (JlA,
0'~)
and (JlB,
O'i).
From the data sets, we
compute (XA,s~) and (XB,s~) and we make the null hypothesis H 0 that the two
population variances are equal
2
2
null hypothesis, H 0 : O'A = O'B
Chap. 7
176
Hypothesis Tests
with the alternative hypothesis HA that the two variances are not equal
2
alternative hypothesis, HA: O"A
* O"B2
The appropriate statistic is
si
where
> s~. We reject Ho if Prob[F > F(IA - 1, IB- 1, a)]> a and accept Ho if
Prob[F > F(IA - 1, I B - 1, a)] ::;;; a.
s1 si
We assume that
>
because F?:: 1 and that simplifies the standard tables. There
is one table for each value of a. A limited portion of a standard table is given in Table
7-1. More extensive tables are given in Abramowitz and Stegun (1965) and many other
statistics books.
Table 7-1. Critical values for the F test with a= 0.05.
JA-1
.............
1
2
3
4
5
6
8
10
20
30
00
161
18.5
10.1
7.71
6.61
5.99
5.32
4.96
4.35
4.17
3.84
200
19.0
9.55
6.94
5.79
5.14
4.46
4.10
3.49
3.32
3.00
216
19.2
9.28
6.59
5.41
4.76
4.07
3.71
3.10
2.92
2.60
225
19.3
9.12
6.39
5.19
4.53
3.84
3.48
2.87
2.69
2.37
230
19.3
9.01
6.26
5.05
4.39
3.69
3.33
2.71
2.53
2.21
234
19.3
8.94
6.16
4.95
4.28
3.58
3.22
2.60
2.42
2.10
239
19.4
8.85
6.04
4.82
4.15
3.44
3.07
2.45
2.27
1.94
242
19.4
8.79
5.96
4.74
4.06
3.35
2.98
2.35
2.16
1.83
248
19.4
8.66
5.80
4.56
3.87
3.15
2.77
2.12
1.93
1.57
250
19.5
8.62
5.75
4.50
3.81
3.08
2.70
2.04
1.84
1.46
254
19.5
8.53
5.63
4.36
3.67
2.93
2.54
1.84
1.62
1.00
In-1
1
2
3
4
5
6
8
10
20
30
00
Statistics for Petroleum Engineers and Geoscientists
i77
Example 1 c - Comparing Two Sample Variances. Let us reconsider the
previous data to compare the sample variances.
Since 1.67 < 1.91, the critical value, H 0 , is accepted. Therefore, the values
=5 and =3 do not indicate that the underlying population variances are
unequal. They may, in fact, be unequal, but the variability in the sample
variances is sufficiently large to mask the difference. Further samples or
other information (e.g., geological provenances of the samples) are needed to
make this assessment.
s1
si
Example 2- Comparing Dykstra-Parsons Coefficients. Consider the case
where two Dykstr~-Parsons coefficients CVvp's) have been estimated from two
wells in a field: VDPl = 0.70 (from I 1 = 31 core-plug samples) and
A
Vvp 2 = 0.80 (from I 2 = 41 core-plug samples). We want to determine
whether the difference in VDP can be explained by sampling variability at the
5% level.
Since VDP is being used to assess variability, we assume that permeability is
log-normally distributed (see Chap. 6). The standard error of Vvp estimates
is given by
sv= -1.49[ln(l-Vvp)](l- Vvp)]!{j
where Vvp is the "true" population value (Jensen and Lake, 1988). Here we
will assume that VDP = 0.75, a value between the two estimates and in
A
A
keeping with our null hypothesis that Vvp 1 and Vvp 2 are not statistically
different. The standard errors for the two estimates are
Chap. 7
178
~v1 =-1.49[1n(1-0.75)](1-0.75)rJ3i
Hypothesis Tests
= 0.093
and
~Vz::::: -1.49[1n(1-0.75)](1-0.75).f'}4i =0.081
· A simple sketch shows that the 95% confidence intervals (of size± 2 standard
errors) for these estimates overlap considerably (Fig. 7-4). Hence, no
statistically significant difference appears to exist.
~DPl
~"
'"
0
Figure 7-4.
•
95%
0.50
0.70
0.80
I
I
I
•
•
0.90
I
95%
~DP2
•
I
.k
Confidence intervals for two Dykstra-Parsons coefficient estimates.
1\
1\
1\
Quantifying this assessment, we have LlVvp =VDPz- Vvp 1 =0.10 and
1\
1\2
1\2
2
2
.
Var(LlVvp) =sv + sv =(0.093) + (0.081) =0.015. Hence, the standard
1
1\
2
- ,-;::-;:::::;
1\
error of LlVDP is about"\' 0.015 = 0.12, giving Ll VDP'= 0.10 ± 0.12, which
is not statistically different from 0. Such large standard errors are typical and
probably account for the lack of differentiation of formations in the work of
1\
Lambert (1981), inasmuch as it is rare to calculate Vvp from more than 100
samples.
There is, however, a problem with the above analysis. Vvp's are not additive
and, therefore, the implicit comparison shown in Fig. 7-4 may not
be.
1\
meaningful. We see, for example, that ~v1 is more than 10% of Vvp 1, and
we may no longer be in the region of local additivity for this level of VDP·
The variances, however, should be additive and can be compared using an F
test.
Statistics for Petroleum Engineers and Geoscientists
179
As shown in Chap. 6, VDP = 1 - e-a, where a2 is the variance of the log/\
permeability PDF. We can estimate a from the Vnp's and compare the
variance estimates.
"2
sy2
F
2.59
=:5Jl = 1.45 = 1.80
The F statistic for this case is F(40, 30, 0.05) = 1.79
Consequently, comparing the variances shows that the difference in the two
variabilities is statistically significant at the 95% level. This contradicts the
assessment based on the VDP standard errors.
The F test for equality of variances has two weaknesses. First, it is more complicated
than the t test because it requires two degrees of freedom and F values from tables. There
are no values readily related to the normal-distribution probabilities as has the t
distribution. Second, it is very sensitive to nonnormality of the populations (Miller,
1986). The F test is much less sensitive to nonnormality, however, when used for
testing equality of means.
When means are being tested, the procedure is termed analysis of variance or ANOVA.
Once the basis for ANOV A is understood, it can be appreciated why it is called an
analysis of variance and not an analysis of means. The ANOV A method uses the F
distribution in the following manner.
Suppose we have samples from three populations N(IJA, a;), N(IJB, a;), and
N(IJc, a~) with
a~= a 2 . The null hypothesis (Ho) is that llA =liB= Pc
with the alternative that either or both equalities are untrue. From the data sets, we
J\2
J\ 2
A 'l.
,
compute (XA, sA), (Xs, sB), and (Xc, sc) for set s1zes IA,lB, and I c. The overall
average is given by
ai =a]=
X
The variability of the averages is
IAXA + IsXB + IcXc
IA+IB+Ic
Chap. 7
180
Hypothesis Tests
where the denominator is 2 because we have three averages with (3 - 1) degrees of
freedom. The total variability of all the data in the sets is
This gives us all we need to apply the F test. The appropriate quantity is
where we reject H0 if Prob{ F > F[2, (!Kl) + (JB-1) + (/c-1), al} >a and accept Ho if
Prob{F > F[2, (/Kl) + (JB-1) + (Ic-1). a] } :-.:;;a.
In this case, F is a ratio comparing two variabilities (hence the name analysis of
variance), both of which are estimates of cP. The numerator is the variability based on
the three averages, XA, XB, and Xc. while the denominator is based on pooling all
the data together. We will use this approach again in regression, for the coefficient of
determination. The above expressions can be extended to tests on many means.
Example 3- Means Testing Using the F test. Let us retest the means using
ANOVA for the data sets from Example 1. Ho: JlA = JlB;HA: JlA JlB·
*
~1 =50
XB = 48
X
:::
25•50 + 30•48
25 + 30
25 (50-48.9)2 + 30 (48-48.9)2
1
=48.9
= 54.5
Statistics for Petroleum Engineers and Geoscientists
181
Since F(l, 53, 0.05) = 4.03, HA is clearly indicated. This is the same result
as using the t test. Indeed, it can be shown that F(l, /, a)= [t(/, a)] 2 .
.
. . to unequa1vanances
.
(e.g., aA2 1:. aB2 1:. ac
2) among
. mo derate1y msens1t1ve
The F test 1s
the populations.· The ANOV A method can be extended well beyond the limited
application shown here. Box et al. (1978) do an excellent job of presenting the various
applications and interpretations of ANOV A.
When it is desired to know more than just that the means are not all equal, a multiple
comparison test is needed. It will tell which mean is greater for a stipulated confidence
level. Box et al. (1978) and Rock (1988) discuss several such tests. We reiterate here,
however, our earlier caution that hypothesis testing addresses only the question of
statistical variability. If the means are found to be significantly different, further analysis
should be undertaken. This analysis would include any geological information available
for the data sets and whether the result of different means has an engineering significance.
Recall our earlier comment (relating to Fig. 7-2) pointing out that statistical significance
may not translate to significant differences in flow performance. Facies geometries and
the flow process may either mitigate large contrasts or magnify statistically insignificant
differences in reservoir properties. See, for example, the Rannoch and Lochaline flood
results in Chap. 13.
7m2
NONPARAMETRIC TESTS
These tests do not assume anything about the underlying distributions of the populations
under investigation. There is a price to be paid, however, for not providing PDF
information in hypothesis tests. Nonparametric tests are not so powerful as parametric
tests if the data actually come from the stipulated distribution. As remarked with the t
and F tests, however, the superiority of parametric tests may quickly vanish (e.g., the F
test for equality of variances) if the true PDF's do not accord with the assumed PDF's.
Chap. 7
182
Hypothesis Tests
Mann-Whitney-Wilcoxon Rank Sum Test
This hypothesis test can go by several other names, including the Mann-Whitney and the
Wilcoxon Rank Sum Test. Strictly speaking, the Mann-Whitney test is slightly different
from the Wilcoxon test, but they can be shown to be equivalent. It is a test that is
sensitive to changes in the median, but it is actually a comparison of the empirical CDF's
of the two data sets. Hence, the test will also respond to changes in other features of the
distribution, including skewness. It is only marginally less powerful than the t-test when
the data are samples from normal populations. It may also be used for ordinal- or
interval-type variables, an important tool for comparing geological (indicator) variables.
Suppose we have two data sets, X 1,X2,X3, ... ,XJ and Y1, Y2, Y3, ... , YJ. with
I :s; J. We assume for the null hypothesis that they come from populations with the
same CDF. We mix the two sets and order the data from smallest to largest. The
smallest datum has rank 1 and the largest has rank I+J. LetR(Xi) be the rank of Xi in
the combined set and Tx be the sum of ranks of all I X's:
I
Tx=
L,R<XD
i=l
If I~ 10 and Ho applies, Txis approximately normally distributed:
T -N(I(I + J + 1) 'IJ(I + J +
X
2
12
l))
Therefore, a single- or double-sided test result can be applied. If 10 ~ J > I, special tables
must be consulted (e.g., Rohatgi, 1984). If ties occur, the rank assigned to each value is
the average of the ranks. Rice (1988, Chap. 11) gives a good account of the theory
behind this test
There are two points that drive the Mann~Whitney test. First,
T
x+
T
y=
/(I + J + 1)
2
+
J(I + J + 1)
2
=
(I + J)(I + J + 1)
·
2
so the ranks must sum to a number that depends only on the data-set sizes, and each data
set should contribute to the sum according to the size of the data set.· It also suggests that
Tx and T y are equivalent statistics, and a test on one will yield the same result as a test
on the other. Second, if Txis particularly large or small compared to the size of the data
Statistics for Petroleum Engineers and Geoscientists
183
set (i.e., the X data are dominating the high or low ranks, respectively), the suggestion is
that the X and Y populations are not the same.
Example 4 - Testing Directional Permeability Data. Suppose that we have
the following core-plug permeability measurements from a massive (i.e.,
structureless) sandstone interval. There is no information about sample
depths. Is there a statistically significant difference in the distributions of the
horizontal and vertical permeabilities? Our H 0 assumes that there is no
statistically significant difference between the distributions against the
alternative (HA), and that there is no preferred direction to the difference.
kh: 411; 857; 996; 1010; 1510; 1840; 1870; 2050; 2200; 2640; 2660; 3180 mD
kv:700; 858;967; 1060; 1091; 1220; 1637;1910;2160;2320;2800;2870 mD
We are testing the medians here because of the structureless nature of the
sediment. Testing the arithmetic or harmonic averages would be inappropriate because of the layered nature of the medium that they imply (see Chap. 5).
By inspection, the medians are kMh = 1855 and kMv = 1428 mD.
Comparing these values with the arithmetic averages,
= 1768 and
f"v = 1633 mD, we see that the differences are less than 15%. This suggests that, despite the flow implications for certain averages, the heterogeneity is sufficiently low that any measure of central tendency would suffice.
The ranks of the combined set are shown in Table 7-2.
fh
For these data, Th = 154 and Tv= 146. Note Th +Tv= (12+12)(12+12+1)/2
= 300, as a quick check of our mathematics. In this case, I= J, so we can
analyze Thor Tv, and both samples are larger than 10, so we can use the
normal approximation.
E(Th):::: 12(12+12+1)/2 = 150 and Var(T~ = (12•12)(12+12+1)/12 = 300
Hence, Z =
Th- E(Th)
--Jvar(T~
= (154- 150)/17.3 = 0.23.
A table of Z values
suggests that there is a 40% chance of observing Th ~ 154 under H0. Hence,
H o is upheld.
Chap. 7
184
Hypothesis Tests
Table 7-2. Permeability data and ranks for Example 4.
Data value
700
857
858
967
996
1010
1060
1091
1220
1510
1637
1840
1870
1910
2050
2160
2200
2320
2640
2660
2800
2870
3180
Data type
Rank
1
2
3
411
h
v
h
4
v
v
5
6
7
8
9
10
11
12
13
h
h
v
v
v
h
v
h
h
14
1
15
16
17
18
19
20
21
22
23
24
v
h
~
v
h
v
h
h
'
v
v
o"
h
If we had more information in Example 4, a more sophisticated analysis could be
performed to look for systematic differences between kv and kh. We might also apply a
one-sided t test, allowing only the alternative kv < kh, if we knew the sediments were not
reworked (e.g., dewatered or bioturbated). If we had depth information and the samples
were paired (i.e., one kv and one kh measurement at each depth), we might have checked
for systematic variations. Core-plug depths, however, can be inaccurate and, without
reference to the core or core photographs to check sampling locations, so-called paired
samples may be taken in quite different locations.
Kolmorogov-Smirnov Test
This hypothesis test may be used to compare the empirical CDF against either a known
CDF or another empirical CDF. The test may be used for nominal, ordinal, or ratio
Statistics for Petroleum Engineers and Geoscientists
185
data. It is based on measuring the maximum vertical separation between the two
distributions (Fig. 7-5). If dmax is small, the CDF's F 1 and F 2 are probably not
statistically different. It is a particularly helpful test because, while it may be easy to
compare averages or medians of distributions, comparing entire CDF's is more involved.
1
F(x)
0
X
Figure 7-5. Comparison of two CDF's in the Kolmogorov-Smirnov test.
Each empirical CDF Fi is based on a data set of size Ii so that each Fi will increase in
steps of size I/ or integer multiples thereof. The producti 1I2dmax will be an integer,
and that is the way the critical tables are listed. For small samples (/i ~ 16), no
approximation is available and tables are needed to interpret the result. Rock (1988)
points out that caution is required with tables because some give the value I 1I2dmax that
must be exceeded while others give the critical value, i.e., the latter give the value
(!1!2dmax + 1). If Ii > 16, Cheeney (1983) gives the 5% two-sided critical value that
must be exceeded as either 1.3611-112, where the comparison is with a theoretical CDF, or
1.36(! 1J 2)112(J 1 + 12)-1/2 for comparing two empirical CDF's. The one-sided 5% test
values use 1.22 instead of 1.36 so that approximately 20% fewer data are needed for a onesided test than for a two-sided test.
Example 5- Comparing Wireline and Core-Plug Data. We have core-plug
( </Jp) and wireline ( </Jw) porosity measurements over the same interval in a
well (Table 7-3). Visual inspection of the core suggests all the data came
from similiar facies. At a 5% level, is there a statistically significant
difference in the distributional forms of the two porosities? We assume H 0
such that there is no statistically significant difference between the forms.
against the alternative HA that they are different.
Chap. 7
186
Hypothesis Tests
Table 7-3. Ordered wireline and core-plug data for Example 5.
tPw
tPp
Zw
z,
32.3
32.2
31.7
31.7
31.6
31.3
31.2
30.9
30.7
30.6
30.6
30.2
30.2
30.1
30.1
30.1
30.0
29.6
29.6
29.1
28.5
28.3
27.9
28.4
28.3
27.8
27.5
27.3
27.3
26.7
26.7
26.6
26.6
26.5
26.5
26.4
26.4
26.3
26.3
26.2
26.2
26.0
26.0
25.8
25.2
23.1
1.61
1.56
1.14
1.14
1.78
1.69
1.21
0.93
0.74
0.74
0.17
0.17
0.07
0.07
-0.02
-0.02
-0.12
-0.12
-0.22
-0.22
-0.31
-0.31
-0.50
-0.50
-0.69
-1.26
-3.26
1.04
0.83
0.73
'0.41
0.31
0.20
0.20
-0.11
-0.11
-0.22
-0.22
-0.22
-0.32
-0.69
-0.69
-1.10
-1.63
-1.73
-2.15
Pi
97.8
93.5
89.1
84.8
80.4
76.1
71.7
67.4
63.0
58.7
54.3
50.0
45.7
41.3
37.0
32.6
28.3
23.9
19.6
15.2
10.9
6.5
2.2
Because of the different volumes of investigation and measurement conditions, the average porosities and variabilities may be different for C/Jw and C/Jp·
We will adjust the data so that they are more nearly comparable. Each data
set has been normalized by subtracting the average from each datum and
dividing by the standard deviation to yield the Z values in Table 7-3.
Subtracting the average adjusts for the difference of measurement conditions:
in-situ for wireline compared to laboratory conditions for plugs. Dividing by
the variability adjusts for the different sampling volumes. This is because we
assume that the formation can be divided into units of constant porosity
(porosity-representative elementary volumes using Bear's (1972) terminology). Both the plug and wireline measurement sample one or more of these
units. We further assume that either measurement is the arithmetic average of
187
Statistics for Petroleum Engineers and Geoscientists
the fonnation porosities within its volume of investigation. Each measurement's variability will decrease as n·l, where n is the number of units of
constant porosity. Hence, dividing each measurement by the apparent
standard deviation of the porosity in the interval will compensate for the
difference in volumes.
The CDF's of the normalized data are plotted in Fig. 7-6 using the probabilities shown in Table 7-3. The maximum distance between the CDF's is
dmax = 67.39- 54.35 = 13.0%, or l1/2dmax = 23•23•0.130 = 69. The 5%
critical value.is 1.36(23•23)112(23+23)•1/2 =5, which is exceeded considerably by fti 2dmax· Hence, the two CDF's are statistically different at the5%
level.
What is surprising about the result of Example 5 is not that the CDF's are
different but where they are different. Inspection of Fig. 7-6 shows that the CDF's
are different at several places; dmax occurs near the arithmetic average of the
porosities, not at regions in the tails, as might be expected. Clearly, the plug and
wireline measurements are not responding similarly and simple shifts or
multiplying factors will not reconcile them.
100
_t_ -
80
60
F(x)
40
......•
I
20
...........f'flli:.!CE---
0
••••••• 1
-4
-3
-2
-1
0
Normalized Porosity·
Figure 7-6.
1
Empirical CDF's for Example 5.
2
188
7-3
Chap. 7
Hypothesis Tests
SUMMARY REMARKS
Hypothesis testing has several uses in the analysis of data. Care must be taken, however,
to ensure that the correct problem is. being posed that hypothesis testing is equipped to
answer. The additivity of variables under test can be a major problem and solutions are
not always apparent. Many statistical-analysis software packages contain large amounts
of hypothesis-testing routines, and itis tempting to apply these to our data. The results,
however,may mislead us unless we have a clear physical basis for interpretation.
Perhaps one important role that hypothesis testing can fulfill is to raise awareness of
the value of auxiliary information. For example, the data requirements for one-sided tests
are often considerably lower than for two-sided tests. The geological characterization of
the sediments under analysis may help restrict the analysis so that more may be learned
from the data
8
I TE A A y IS:
ELATI
Correlation is the study of how two or more random variables are related. It is a much
less powerfultechnique than regression, discussed in Chaps. 9 and 10, because correlation
does not develop a predictive relationship. It can be useful, however, for establishing the
strength and nature of the association, or covariation, between random variables.
Since correlation centers on random variables, it is unsuitable for relationships
involving deterministic (or "controllable") variables. For example, in many bivariate
problems, one of the variables can be considered as a quantity under our control with little
or no error (such as a core flood where the amount of fluid injected is a controllable
variable). In other situations, however, we simply measure both the X andY values on a
number of samples with no control over either X or Y, e.g., porosity and permeability of
core plugs. In these latter situations, correlation analysis is appropriate.
8-1
JOINT DISTRIBUTIONS
We have seen in Chap. 3 that all random variables have a distribution function that
characterizes their probabilities of occurrence. If we have two random variables, X and Y,
their probabilities of occurrence will be expressed through their joint probability density
function,f(x, y):
189
Chap. 8 Bivariate Analysis: Correlation
190
Prob(x
<X~
x + OX andy < Y ~ y + oy) = f(x, y) ox oy
If X and Yare independent, a knowledge of X or Y has no effect upon the value of the
other variable and the joint PDF is simply the product of the two separate PDF's:
f(x, y) = fx(x)fy(y)
since Prob(x <X~ x + 8x andy< Y ~ y + 8y) = Prob(x <X~ x + 8x) Prob(y < Y ~
+ oy). In this case, as shown in Chap. 4, E(XY) = E(X)E(Y).
y
The joint PDF of X and Y is an extension of the univariate situation where X and Y
each has its own PDF./x andfy. X andY may both be continuous, discrete, or mixed
with X of one type and Y of another type of variable (see Chap. 3 for a description of
these variable types). Plottingf(x, y) requires three axes (Fig. 8-1). The projection of
f(x, y) onto the x andy axes produces the marginal distributions of X andY. These are
the PDF'sfx andfy that we observe when we ignore the other variable. The areas under
each of these distributions must be one to ensure that they represent probabilities.
f(x, y)
-
Figure 8-1.
X=Xz
A mixed joint PDF for X discrete andY continuous, with their
bimodal marginal PDF's.
Several features of the X-Y relationship can be obtained from the joint PDF. If one of
the two variables is fixed within a small range, say x 0 <X~ x 0 + ox, while Y is allowed to vary, we can obtain the conditional PDF of YJYix(y I X= x0). Geometrically,
fyiX(y I X= xo) is obtained when the plane X= xo "slices through" f(x, y). The conditional PDF can be used to determine how the mean value of Y varies with x0:
Statistics for Petroleum Engineers and Geoscientists
i 91
+oo
E(Y I X= x 0) =
f
y /y 1x(Y I X = x 0 ) d y
E(Y I X= x0) is called the conditional expectation of Y and is a function of x0. There is
a similar conditional PDF and expectation of X. The joint PDF is also used to measure
the X-Y relationship strength, which we discuss further below.
Just as with the univariate case, there are also joint CDF's. Conditional CDF's are
obtained from the joint CDF by holding one of the variables constant. Recall that in
Chap. 3 (Example 3), we used conditional CDF's to relate the sizes of shales to their
depositional environments. In that case, the depositional environment and shale size are
the two random variables. The depositional environment variable has a strong effect on
the shape of the shale-size CDF.
A plane can be drawn parallel to the x-y axes such that f(x, y) is constant (the
"tombstone" sketch of Fig. 8-2). The plane represents the probability p, where
p = Prob( Yl < Y:::; Y2• X= x2) + Prob( Y3 < Y:::; Y4· X= x1)
f(x, y)
Figure 8-2.
X
A plane with constantf(x, y) represents (X, Y) pairs with a
specific probability of occurrence.
Since probabilities are areas under the PDF, p can be interpreted as the following sum of
areas:
Chap. 8 Bivariate Analysis: Correlation
192
p
=
Y2
Y4
Y==yl
Y==y3
Jj(x2,y)dy + Jf(xloy)dy
In a common case where both X and Yare continuous variables and the joint PDF is
unimodal,f{x, y) =c will be a single closed curve, often elliptical in shape. We will see
examples of this below.
8-2
THE COVARIANCE OF X AND Y
How do we measure the strength of covariation between the random variables X and Y?
One such measure of this relationship is the covariance of X and. Y, defmed as
+oo +oo
Cov(X, Y)
=E[(X- f.lx)(Y- f.ly)] =
J
J<x-f.lx){y-f.J,y)f(x, y) dx dy
(8-1)
where E(X) = JlX and E(Y) = f.lY· The covariance can also be written as
Cov(X, Y) = E(XY) - /lx /ly
from the properties of the expectation operator (Chap. 4). The covariance is one of the
most important devices in all of multivariable statistics; it will be used extensively in the
remainder of this text, so it is fitting to devote some space to its properties.
The covariance is a generalized variance. That is,
Cov (X, X)= Var(X)
from its definition and that of the variance. Thus, we would expect Cov and Var to share
many of the same properties. As we shall see, this is only partially true, since the added
generality makes Cov substantially different from Var except in the degenerate case
mentioned above. For example, whereas we must have Var(X)::?! 0, Cov can be either
negative or positive. Additional properties are
covariance is commutative:
Cov(X, Y)
= Cov(Y, X)
from the definition (Eq. (8-1)), and
covariance with a constant cis zero: Cov(X, c) = 0.
This latter property is reasonable once it is understood that Cov measures the extent to
which the first argument varies (covaries) with the second. Any fluctuations in the
random variable X will not have any like variations with a constant.
Statistics for Petroleum Engineers and Geoscientists
193
The reader might notice that, although the covariance was introduced and continues to
be based on the notion of the joint PDF, we begin to use it without showing the PDF
explicitly. This practice will continue to the end of the text. It is a good idea, however,
to keep the joint PDF and the associated probability defmitions in mind, since these form
the basic link between probability and statistics.
One manifestation ofthis is the covariance with multiple random variables Xi and Yj=
Cov(± aiXi,
1=1
i
bjYj)=
J=l
If
aibjCov(Xi, Yj)
i=l j=l
which is a relationship that occurs in several applications.
The most important property of the covariance, however, lies in its ability to measure
the association between two random variables. If X and Yare independent, then Cov(X,
Y) = 0, a property that follows from the independence property of the joint PDF given
above. The converse, however, is not necessarily true; if Cov(X, Y) 0, then X and Y
might not be independent. Meyer(1966, pp. 144-145) gives an example.
=
To form a useful baSis of comparison, the measure of relationship strength should be
dimensionless. The covariance of X and Y is often normalized by dividing by the
variabilities of X and Y. The resulting quantity is the correlation coefficient of X and Y,
given by
p()(, Y)
=
Cov(X, Y)
..J Var(X) Var(Y)
From the properties of the expectation (Chap. 4), this equation can be rewritten as
p(X, Y)
or
=E[(~~:~x~~;~)J
p(X, Y)
=E(X*Y*)
where x* = (X - J.lx)tV V ar(X) and y* = (Y - J.ly)!.Y Var(Y). Since x* and y* are standard
random variables with zero mean and unit variance, p(X, Y) can be viewed as the mean
value of the product of two standardized variables. Standardization shifts the center and
scale of a cloud of (X, Y) values to the origin with equal scales on both axes (Fig. 8-3).
Each point (X*, r*) will lie in one of the four quadrants formed by the axes (Fig. 83). In quadrants 1 and 3, the product x*y* is positive, while x*y* is negative in
Chap. 8 Bivariate Analysis: Correlation
194
quadrants 2 and 4. Thus the sign of p(X, Y) suggests, taking all possible pairs (X, Y),
whether the relationship is mostly in quadrants 1 and 3 or quadrants 2 and 4. The
narrower the "cloud" is around either the 45" or 135' lines, the larger p or -p, respectively,
will be.
y*
Quadrant2
E(Y)
E(X)
X
X*
Quadrant4
Figure 8-3.
Standardized variables x* andy* shift the original X-Y
relationship (left) to the origin and produce a change of scales.
The shaded region represents all (X, Y) pairs within a given
probability of occurrence.
Some properties of the correlation coefficient include
1. lp(X, Y)l :0:::: 1 and
2. lp(X, Y)l = 1 if and only if Y =aX+ b for some constants a and b, a:tO.
Although p is not in general a good indicator of relationship strength, it is better at
recognizing linear interdependence as indicated by property 2. Even so, the correlation
coefficient should be used with caution. The behavior of p(X, Y) depends on the joint
distribution of X and Y, so what may work well with one set of data may not work well
in another case, as shown below.
Example 1 - Comparing Correlation Coefficients. Consider the case of three
variables, X, Y, and Z, where we have 0 < lp(X, Y)l < lp(X, Z)l < 1
(Fig. 8-4). Do X and Z have a stronger linear element to their relationship
than do X and Y?
The X-Y figure (left) has the same Xi coordinates as the X-Z figure for
-1 <X< 1. Thus Y = Z for -1 <X< 1. Z = pX + (1 - p2)112t:, where X
and e- N(O, 1), making X and Z joint normally distributed (Morgan, 1984,
Statistics for Petroleum Engineers and Geoscientists
195
p. 89). For this distribution, the Z- X covariation is strictly linear (see
below). Thus, both data sets have identical linear elements to their
relationships, while the estimated p's are different.
4
y
4
2
2
z
0
0
-2
-2
p =0.74
p =0.90
+
-4
-4
-3
-2
0
-1
2
1
-3
3
-2
-1
1
2
3
X
X
Figure 8-4.
8E3
0
Bivariate data sets for Example 1.
THE JOINT NORMAL DISTRIBUTION
One case for which the correlation coefficient p is important is if X and Yare joint
normally distributed (or bivariate normally distributed). If X and Yare joint normally
distributed (JND), then
fir,x, y) =
where
2ncrxay·h - p 2
exp [ x
*2
- 2px *y * + y * 2]
2
-2(1 - P )
(8-2)
x * ::::: ~
Cf)(
with .ux = E(X) and crx2 = V ar(X)
y*
and
1
=Jl...::...1!L
cry with ,uy = E(Y) and cry2 = Var(Y)
crxy
crxcry
p = - - with
axy = Cov(X, Y)
p is the correlation coefficient between Y and X and is a population parameter of the
distribution, just as J.l.x and ).l.y are.
Chap. 8 Bivariate Analysis: Correlation
196
The joint normal PDF looks like Fig. 8-5 when p = 0.7. When p = 0, a plane
= constant makes a circle; when p -:t= 0, the form is an ellipse. The closer Ipi is
to 1, the more elongated the ellipse becomes. The major axis of the ellipse is at 45' or
135' to the X-Y axes. The interior of any ellipse represents those (X, Y) pairs having a
given probability of occurrence. Hence, each ellipse has a probability level associated
with it. This probability is given by the volume of the PDF within the boundary of the
ellipse (recall Fig. 8-2 and associated discussion). Formulas for the ellipses and their
associated probabilities are given in Abramowitz and Stegun (1965, p. 940).
f(x *, y*)
f(x*, y*)
x* = 0
x* = +1
x*
= +2
Figure 8-5.
A joint normal PDF with p = 0.7. The
marginal PDF's of x* andy* are both N(O, 1).
The JND ellipses are often plotted with data on scatter plots. The ellipses are a
diagnostic tool to help detect significant deviations from joint normality. Drawing such
an ellipse does not imply that the data come from a JND population, however. For
example, if 5 out of 40 points fall outside the 95% ellipse, it is rather unlikely that the
data are JND. A test for X andY JND is to examine the marginal PDF's; if either X or Y
is not normal, they cannot be JND. Marginal normality is necessary but not sufficient,
however. Hald (1952, pp. 602-606) discusses a test (the 2 test) for joint normality.
z
Example 2 -Assessing Joint Normality with Probability Ellipses. Figure
8-6 shows two 100-point data sets with their 90% probability ellipses. As in
Example 1, we have constructed theY data from Y = pX + (1- p2)112e,
where X and e- N(O, 1) and p = 0.9. For Z, however, we have
Statistics for Petroleum Engineers and Geoscientists
197
Z = pX + 3.46 (l-p2)1128, where 8 is a uniformly distributed variable with
upper and lower endpoints of 0.5 and -0.5, respectively. The factor 3.46 is to
keep Var(8) = Var(e) = 1. X andY are JND, but X and Z are not. Both Y
and Z have marginal distributions very close to normal, as judged by
probability plots.
When comparing distributions, differences are usually most pronounced at the
extremes, rather than near the middle of PDF's. Therefore, we choose a large
probability ellipse (e.g., 90, 95, or 99%) and examine the behavior of the
points falling outside this ellipse. The more data we have, the larger
probability level we can choose. In this case with 100 data, a 90 or 95%
level is appropriate, and we have chosen the 90% level (Fig. 8-6).
In line with the probability level, about ten points lie outside the ellipse for
the JND data (Fig. 8-6 left). These points are not clustered at one area and are
near the ellipse. In contrast, the non-JND data set (Fig. 8-6 right) has fewer
data outside the ellipse, the deviations are concentrated at the extreme upper
and lower parts of the ellipse, and there is one point (X= -2.5, Y = -2.9) well
away from it. Such differences might give cause to resort to more formal
tests (such as the x2 test) to determine the joint normality of X and Z. In
this case, the non-JND data do fail the test.
3
2
y
1
0
-1
-2
+
-3
-3
-2
-1
0
X
Figure 8-6.
1
2
3
-3
-2
-1
0
1
X
Two data-set scatter plots with normal probability ellipses.
2
3
Chap. 8 Bivariate Analysis: Correlation
198
When X andY are JND, several nice properties apply to X andY (see, for example,
Hald, 1952, pp. 590-601, for a detailed discussion with figures):
1.
2.
3.
4.
The conditional PDF's of X and Y have a normal (Gaussian) shape, but these
shapes do not have unit area. (The volume under the full PDF is equal to one.)
This is plausible from inspection of Eq. (8-2) and Fig. 8-5.
The marginal PDF's of X and Y are both normal. To obtain this result, we
integrate Eq. (8-2) over all values of one variable.
The standardized variable x*- r* covariation is linear, along a 45• or 135• line.
The conditional means are linear functions of Yo and x0:
ax
E(X I Y =Yo) = J.Lx + p - (yo- )ly) for variable x and fixed y
ay
(8-3a)
O'y
E(Y I X = x 0) = J.Ly + ~ (xo - J.Lx) for variable y and fixed x
(8-3b)
and
ax
These are the equations for regression lines in the X and Y JND. Thus, a linear
regression model (Chap. 9) is automatically appropriate.
An estimator for pis the sample correlation coefficient, given by
where S is the sum of squares of the products shown in its subscripts and the summations
are over the I data. I-lsAB is an estimate for aAB• where A andB represent X or Y. r2
is equal in value to the coefficient of determination (R 2) when X andY are JND. R2,
however, has a different interpretation from that of r, which is an estimate of a population
parameter when X andY are JND. Otherwise, r can be used as a measure of the linear
interdependence of X and Y. R2 has an interpretation no matter what the joint PDF of X
andY is or even if X is deterministic, not random. R2, however, must have a model
(which might not be linear) to indicate what the anticipated form of the covariation is.
R2 will be discussed further in Section 9-9.
There are significance tests available for r. The PDF of r is rather complicated, but the
transformed quantity (1/2) [ln(l + r) - ln(l - r)] is approximately normal with mean
(1/2) [ln(l+ p) -ln(l - p)] + p/[2(I- 1)] and variance 1/(I- 3) for I data (Hald, 1952, p.
609). Tables are widely available (e.g., Snedecor and Cochran, 1980, pp. 477-478). A
common test is to determine whether lrl is sufficiently large for the sample to indicate that
pis significantly different from zero. These tests, however, are only applicable if X and
Statistics for Petroleum Engineers and Geoscientists
199
Yare JND (or nearly so). Variabilities and significance tests for X, Y, SXY, Sxx. and
Syyare fairly complicated and given in Anderson (1984, Chaps. 4 and 7).
The estimated slopes of the regression lines are given by
fSYY and r \Jsyy
- rsxx
\JSXi
r -
for Y upon X and X upon Y, respectively. We can calculate r without calculating any
regression lines, since it requires neither the slope nor the intercept of either line. This
points out a fundamental distinction between correlation and regression. Correlation
indicates only the presence of an association between variables and has no predictive
power. Regression permits the quantification of relationships between variables (the
appropriate model being known) and their exploitation for prediction.
8-4
THE REDUCED MAJOR AXIS (RMA) AND
REGRESSION LINES
The line at 45o or 135° in the x* "y* plane (Fig. 8-3) represents the form of the X-Y
covariation if X and Yare approximately JND. This line, the major axis of all ellipses
formed from fix*, y*) =constant (Fig. 8-5), is called the line of organic correlation or the
reduced major axis line (see Sokal and Rohlf, 1981, p. 550 for other names). It is related
to the major axis line but has the advantage of being scale invariant (Agterberg, 1974,
pp. 119-123). The RMA line equation is
(8-4)
and is similar in form, but with a different slope, to the regression lines of Eqs. (8-3).
The RMA line lies between the_twQ__regression lines (see Fig. 8-7). All the estimated
lines pass through the point ( X, Y) in the X-Y plane. The RMA line represents the
relationship such that one standard deviation change in X (i.e., o-x) is associated with a
unit standard deviation change in Y (i.e., O'y). When perfect linear correlation exists, all
three lines are coincident. The RMA line is insensitive to the strength of correlation
between X and Y since its slope does not change as O'XY• the covariance of X and Y,
changes.
The RMA line represents how X andY covary; as X changes by o-x, Y tends to
change by O'y. This gives the slope Syy/Sxx for the estimated line. The "tightness"
of the X-Y covariation is estimated by r. There is no suggestion in this relationship that
Chap. 8 Bivariate Analysis: Correlation
200
one variable is a predictor while the other is a predicted variable. The RMA line is
usually more appropriate for applications where prediction is not involved. Such
applications include estimating an underlying relationship or comparing a fitted line to a
theoretical line. For example, suppose we had porosity and permeability data from core
plugs coming from two different lithofacies. To compare the porosity-permeability
relationships of the two lithofacies, the RMA line would be an appropriate choice. See
Example 3 below.
Y-on-X
RMA
X-on-Y
Figure 8-7.
RMA
X-on-Y
JND ellipses of X and Y and their RMA and regression lines.
The two regression lines also define linear relationships between X and Y. The
predictor-predicted roles of the variables, however, are inherent in the lines. The Y-on-X
line, representing E(Y I X = xo), shows how the mean value of Y varies when X is fixed
at the value x 0 . Thus X must be known in order to establish the mean of Y. As the
correlation diminishes to zero, the regression lines become coincident with the axes,
suggesting that the best estimate of X or Y is E(X) or E(Y), respectively. In this case, a
knowledge of one variable gives little help in predicting the value of the other. The
RMA line, however, remains the same and is always coincident with the major axis of
the ellipse.
A geometrical interpretation exists for calculating the RMA and regression lines for a
standardized (X*, r*) data set (Agterberg, 1974, Chap. 4). The Y-on-X regression line is
obtained by minimizing the sum of the squared vertical distances (line ad in Fig. 8-8)
from the line to the data points. The X-on-Y regression line is given by minimizing the
sum of the squared horizontal distances (line ab in Fig. 8-8). The RMA line obtains
from minimizing the sum of the squared distances from the data points to the line ac in
Statistics for Petroleum Engineers and Geoscientists
201
Fig. 8-8. Because of these geometrical interpretations, it is sometimes argued that the
RMA line is the more appropriate line for prediction when both variables are subject to
substantial error. The relative merits of these lines for prediction will be discussed in
Chap. 10.
X*
Figure 8-8.
Different regression lines can be determined from a data set,
depending upon which distances are minimized.
For X andY JND, the variability of the RMA line slope is the same as the Y-on-X
line slope variability, which to order J-1 is (Teissier, 1948)
2
Syy }cry~
Var ( S
2
I
XX
(8-5)
CTx
The RMA Y-intercept variability is (Kermack and Haldane, 1950)
_
a
_
cry
2
[
Var(Y-_rX)=I(l-p)2+
crx
J.J.x(1
+ p)]
2
G
2
X
Kermack and Haldane (1950) also give results for X andY not JND.
Example 3 -Regression and RMA Lines for Core Data. Figure 8-9 shows a
scatter plot for two sets of core-plug porosity (base-10 log) and permeability
data. Both sets are from the Rannoch formation (see Example 6, Chap. 3 for
Chap. 8 Bivariate Analysis: Correlation
202
more details of the Rannach) and come from similar facies within one well
from each field. Field A, however, has a slightly coarser sediment with less
mica than Field B. As expected, the average porosity and permeability of
Field B are less than those of Field A.
All marginal distributions appear approximately normal on probability plots,
and slightly fewer than the expected number of data fall outside the 90%
ellipses: 22 data for A and 5 data forB. The cluster of A data near X= 21
and Y = 1. 7 suggests that these data may not be from a JND population but
are reasonably close for our purposes.
Figure 8-9 also shows the RMA and regression (Y-on-X) lines for both data
sets. The RMA and Y-on-X lines are close for Field B, reflecting the large
correlation coefficient, rB = 0.97. Field A has a smaller r, r A= 0.84, so the
lines are more distinct. Both r's are significantly different from zero (no
correlation). For example, for Field A, O.S[ln(l + rA) - ln(l - rA)] = 2.4 with
standard error(/- 3)-1/2 = 0.06.
3.5
,.-....
+ Field A, 286 data
3.0
x Field B, 74 data
~
.._., 2.5 · - Y-on-X line
eJ:)
·-·-- 1.5
0
~
2.0
,Q
e;:l
~
e
~
<lJ
~
1.0
0.5
lilt
0.0
15.0
Figure 8-9.
20.0
25.0
Porosity, %
30.0
Porosity (X)-permeability (Y) scatter plot and lines for
Example 3.
Statistics for Petroleum Engineers and Geoscientists
203
The RMA lines have different intercepts because of the textural differences of
the two fields. The RMA line slopes are statistically similar, however:
0.162±0.005 for Field A and 0.161±0.005 for Field B. (The standard errors
are from Eq. (8-5).) This suggests that the same mechanism may be
controlling the permeability-porosity covariation in both wells. A different
conclusion might be reached, however, using the Y-on-X lines: Field A has
slope 0.135±0.005 while Field B has slope 0.156±0.005. The regressionline slopes are related to the strength of correlation, which reflects the amount
of variability within. the relationship as well as the nature of the relationship.
For the purposes of comparing relationships, the RMA lines are the easier to
interpret.
The similarity of the permeability-porosity relationships of Fields A and B in
Example 3 suggests that Field B does not have lower porosity and permeability from
causes other than the grain size and mica. We would, therefore, expect that porosity and
permeability estimates from wireline or other measurements could be similarly interpreted
in both fields.
8=5
SUMMARY REMARKS
Correlation analysis provides several procedures for assessing JOint vanatwn
(correlation). Most of these methods are best applied when the data are from a joint
normally distributed sample space because the statistics have a particularly helpful
interpretation and the sampling errors can be evaluated. We will see correlation measures
developed and extended further in Chap. 11, where sampling locations are also involved.
9
BIVARIATE ANALYSIS:
LINEAR REGRESSION
Probably no other statistical procedure is used and abused as much as regression. The
popularity of regression stems from its purpose, to produce a model that will predict
some property from the measurements of other properties, from its generality, and from
its wide availability on calculators and computer software. Regression is especially
useful for reservoir description.
Because reservoirs are below the Earth's surface, measurements made in situ are often
restricted in type and number. Therefore, it is desirable to exploit any relationship
between a property that can be measured and another that is needed but cannot be
measured. It is, however, open to abuse because of numerous aspects of which the user
need not be aware in order to apply the results. It may be only after the unsatisfactory
predictions are applied that a problem becomes apparent.
For two reasons, we will concentrate here on linear regression with one predictor
variable. The first reason is clarity. The conclusions for the bivariate and multivariate
cases are similar, but the matrix expressions can obscure the underlying concepts as more
variables are introduced. The second reason is par$imony, a term borrowed from timeseries analysis, which means that the simplest model tbat explains a response should be
used. Often, the bivariate model will meet this requirement. More elaborate models may.
fit the data better, but complications may arise when, unwittingly, the model is used
outside the range of the data on which it was developed.
205
206
Chap. 9 Bivariate Analysis: Linear Regression
We begin by covering the basic features of least-squares bivariate regression.
Mathematically, the procedure is straightforward and can be covered in a few lines.
Statistically, however, regression has a number of aspects that we consider in detail to get
the most benefit from the data collected. Because of the amount of material involved,
more advanced considerations are deferred to Chap. 10.
9=1
THE ElEMENTS OF REGRESSION
The regression procedure consists of the following elements:
1.
2.
3.
4.
A model (derived or assumed), an equation that relates one or more observed
quantities to a quantity to be predicted. The model contains unknown parameters
that must be evaluated.
Measurements of both observed and predicted quantities.
A method to reverse the role of the variables and the parameters in the model to
determine the latter, based on the measurements.
Application of the model to predict the desired quantity.
Regression is based on using data to determine some unknown parameters in a model.
Hence, a model must exist or be developed (element 1) for the regression to proceed. The
model represents the a priori information we bring to the data analysis, but it is
incomplete by itself. Unknown parameters in the model must be determined from
measurements (element 2) of all the quantities involved. The regression centers on
element 3, in which the model parameters can be determined from statistical arguments,
provided that the data satisfy certain conditions. The procedure has great practical use
because it provides measures to indicate the appropriateness of the model and it provides
parameter values. It does not, however, of itself determine the model.
Element 4 in the above list is also very important. If we are developing a model for a
purpose other than prediction, another procedure may be more appropriate than regression
(e.g., the RMA line discussed in Chap. 8). Recall from Chap. 8 that regression implies
that we want to model how the mean value (i.e., the expected value) of the predicted
quantity varies with the observed quantities.·
Element 1 might be unnecessary in some situations. Consider the plots in Fig. 9-1,
which have a large number of measurements for X andY, two variables of interest. It is
apparent from the left diagram that there is some relationship between the two variables.
Hence, a knowledge of one could be profitably used to predict the value of the other.
Assume that we wish to predict Y from measurements of X. There are enough data that
we could divide up the "cloud" into sections or bins (right figure) and calculate the
arithmetic average of the Y's in each section. This procedure would give us a set of Yj's
Statistics for Petroleum Engineers and Geoscientists
for the i = 1, 2, .. .,I sections.
207
W~ co~d
then use this set to predict Y from X: when
The coarseness of the approximations thus
depends on how many data we have; if we have many data, the sectioning could be quite
fine and we would have a good approximation to how Y varies with X, the regression
relationship, without recourse to a model.
X i-l ~X ~ Xi for some i, then Y
= Yi.
y
y
+
.l
X
X
Figure 9-1.
Many data in an X-Y scatter plot (left) provide directly for
an X-Y relationship.
1\
Of course, because this procedure involves step changes in Y, it requires a considerable
amount of data to develop. It1\also should
not be used outside the measured data range; the
procedure would require that Y = Y 1 for any X > X b no matter how large X is, and it
ignores any information we have about the X-Y relationship. Instead, we may wish to
introduce an equation (element 1) that will tell us, in a continuous manner, how Y
varies with X. Therefore, we want to use prior knowledge (the form of the equation) to
replace the need for so many data and obtain a continuous regression relation.
9~2
THE LINEAR MODEL
The most common type of regression is linear regression. Furthermore, of all possible
models, the linear model is the most frequently used:
Y=
f3o + /31 X + e
(9-1)
where X is the predictor or explanatory variable, Y is the response, e is a random error,
and f3o and /31 are the regression model parameters. X is called an explanatory variable
Chap. 9 Bivariate Analysis: linear Regression
208
because a knowledge of how X varies will be used to explain variations in Y. f31 is also
called the regression coefficient.
Note that "linear" can be used in two ways when considering regression. The first way
describes how the unknowns ([3 0 and /3 1 in the above equation) appear in the problem.
The second way describes how the predictor variable appears in the problem. For
example, the model Y = f3o + fhX2 + e is still a linear regression problem, since the
unknown f3's appear linearly in the model, but the model is nonlinear in X, the predictor.
Since e is a random variable in the model, Y is also a random variable. X might or
might not be a random variable. See Graybill (1961, Chap. 5) for an excellent discussion
of the various situations Eq. (9-1) might represent.
If the model, Eq. (9-1), is describing the data behavior well, the errors are random.
This means that they are independent: individual deviations do not depend on other
deviations or on the data values. It is also common to assume that errors are normally
distributed with zero mean (E(e) = 0). The term "error" here covers a multitude of sins.
It includes measurement errors in the observed values of Y and inadequacies in the model.
For example, e includes the effects of variables that influence the response but that were
not measured.
Model inadequacy is often a significant contributor to the "error." Typically, what we
set out to measure is measured with reasonably good accuracy. The properties that we
cannot measure, coupled with the use of simplistic models, usually account for most of
the prediction error. This aspect of the error is often overlooked in discussions of its role
and magnitude. However, even if we cannot measure the predictor accurately, regression
is usually the appropriate method when we want to develop a model for prediction.
We can take the conditional expectation of both sides ofEq. (9-1) to give
E(Y I X= Xo) = f3o + fJ1 Xo
since we assume that E(e)
=0.
Likewise, the variance is
Var(Y I X= Xo)
= Var(f3o + /31 X+ e I X= X0) = ae2
We are using conditional operators here because, by assumption, Y depends on X.
Recall that, from Chap. 8, E(Y I X= Xo) is the mean value for Y when X takes the
value Xo and involves the conditional PDF of Y, fyiX(Y I X= X0). The last equation
yields the variance of errors because the errors are assumed independent of X.
209
Statistics for Petroleum Engineers and Geoscientists
9-3
THE LEAST=SQUARES METHOD
We use the method of least squares to estimate the model parameters f3o and [3 1 in
Eq. (9-1) from a set of paired data [(X 1, Y1), (X2, Y2), ... ,(Xb YJ)]. The least-squares
criterion is a minimization of the sum of the/\ squared differences between the observed
responses, hand the ;redicted responses, Yi, for each fixed value of Xi (Fig. 9-2).
These differences, Yi- Yi> are called residuals.
y
X
Figure 9-2. The least-squares procedure determines a line that minimizes
the sum of all the squared differences, (Yi- Yj)2, between the
data points and the line.
In mathematical terms, the preceding description can be expressed as
I
S(f3o,/h)=
~ (ri- ¥)
1=1
I
2=
L
(Yi- f3o- f31Xi) 2
(9-2)
i=l
Equation (9-2) is I times the estimate ofVar(Yi- Yi), since Var(Yi- Yi) = E[(Yi- Yi)2]
1\
and E(Yi- Yj) = 0. Since Sis proportional to a variance, it is always nonnegative.
We would like to find the values of the slope (/31) and intercept (f3o) that makeS as
small as possible (Fig. 9-3).
Chap. 9 Bivariate Analysis: Linear Regression
210
s
min S
Figure 9-3. The least-squares procedure finds the minimum value of S.
A differential change inS, dS, can be written in terms of differential changes in f3o and
{31 as
as
as
dS = CJfJo dfJo + CJ/3 1 d/31
This is a local condition since it applies only to small changes in the variables. At the
minimal S, we must have dS = 0. Since the slope and intercept are independent (i.e.,
?.odf3o + ?.1d/31 = 0 only for ?.a = ?.1 = 0), this can only be true if each partial derivative
is zero.
CJS
From af3o = 0:
as
From af3l
= 0:
(9-3a)
or
or
L (Y·I
Y·) X·= 0
I
I
(9-3b)
where the summations
are1\ from i = 1 to i =I. These equations are the least-squares
/',
normal equations. fJo and fi1 are xhe valuxs of f3o and /31 that minimizeS, but they also
have a statistical interpretation. f3o and /31 are the slope and intercept estimates derived
211
Statistics for Petroleum Engineers and Geoscientists
from a sample of I data; they may differ in value from the Ropulat~on slope and intercept
values. Because the Yi are samples of a random variable, f3o and /31 are also samples of
random variables. Similarly, S is a random variable, related to the variability of the
residuals.
1\
The version of Eq. (9-3b) on the right makes it apparent that the residuals Yi- Yi and
Xi are orthogonal (i.e., have zero estimated covariance). This means that the least-squares
procedure is giving residuals that have no further relationship with X. Thus, the model
and least-squares estimator have "squeezed out" all the explanatory power of X (see Box et
al., 1978, Chap. 14, for further details).
Both of the normal equations are linear; hence, they can be solved as
(9-4a)
and
(9-4b)
to give
the sample regression of Y on X. Equation (9-4b) can also be written as
b
fh = SSxy I SSx, where SSxy represents the sum L(Xi- X)Yi and SSx is the sum of
the squares L(Xi- X)2.
1\
T~e
form of Eq. (9-2) ensures that, within the constraints of the model, the values [3 0
and [3 1 will approximate the mean value of Y at any given X= x. Recall that, from
Chap. 4, the mean value ~inimj_zes the variance. Since S is a measure of the variance of
the data about the line, {3 0 + 1x thus approximates E(Y I X= x). Equation (9-4a)
reflects this property in that f3o is computed to ensure that the line includes the point
J3
( X,
Y).
Example la- Regression Using a Small Data Set. Compute the least-squares
1\
A
estimates [3 0 and [3 1 for the model Y = [30 + [31X + e with the data set
((X, Y)} = {(0, 0), (1, 1), (1, 3), (2, 4)).
Chap. 9 Bivariate Analysis: Linear Regression
212
2
We have 'LXi = 4, 'LYi = 8, 'LXiYi = 12, and LXi = 6. From Eqs. (9-4),
A
12 -
/31 =
1
4 4•8
1
4
6- -4 2
4
-2 = 2
A
and
1
1
f3o =-4 • 8 - 2 • -4 • 4 = 0
A similar approach can be taken with models having more than two unknowns,
although the expressions are best handled using matrix notation. Suppose the model is
A
where we have J explanatory variables and a set of I data. Let Y and Y be the I x 1
vectors of predicted and actual responses, X is the I x (J + 1) matrix of explanatory
=X~ thus
values, and is a (J + 1) x 1 vector of the estimated model parameters.
represents the following equation:
ft
Y
1
x?) x} 2 )
xU)
1
xil) xi2)
xU)
1
2
X
1
Using this shorthand notation, the normal equations, Eqs. (9-3), can be written as
Y) = 0 or, since = X~. the normal equations become xT(Y - X ffi) = 0, where
xT is the transpose of X. If we solve the normal equations for
we obtain
xT (Y -
Y
ft,
Example 1 b- Regression Using a Small Data Set. We re-solve the previous
problem using matrix notation. In this case, J = 1,
Statistics for Petroleum Engineers and Geoscientists
Hence,
~TX]-1 ={[ ~ : : ~ J [
213
!I]r r
=[: :
=[
'[~ -.;; J
and
~] m= [~]
This is the same result as obtained in Example la.
The approach embodied by the above equations, which assume that all the data (Xi, Yi)
are equally reliable, is called ordinary least squares (OLS). The equations for the more
general case where different points have differing reliabilities (weighted least squares, or
WLS) is treated in Chap. 10. For the linear OLS model, the more sophisticated hand
calculators have these equations already programmed as function keys.
9u4
PROPERTIES OF SlOPE AND INTERCEPT
ESTIMATES
Understanding the origin of variances in the slope and intercept estimates takes some
doing, because we have been treating them as single-valued parameters used to describe a
set of data. Recalling the probabilistic approach described in Chap. 2, however, the
observations Yi and Xi are really only the results of experiments from two sample spaces
that are correlated. If we were to pick another set of (Xi> Yi), we would get another slope
and intercept. If we pick many sets of (Xi, Yi), we get a set of slopes and intercepts that
are distributed with mean and variance as given below.
When E(e) = 0 and Var(e) =a;, we can estimate the bias and precision of the slope
and intercept obtained by OLS. See Montgomery and Peck (1982, Chap. 2), Box et al.
1\
(1978, Chap. 14), or Rice (1988, Chap. 14) for derivations of these results. Both f3o and
1\
f31 are unbiased:
214
· Chap. 9 Bivariate Analysis: Linear Regression
More importantly, the precision of the estimates is given by
(9-Sa)
arxl
(9-Sb)
These assume that the Ei values are independent and we know the value of a~.
Usually, we have to depend on the data to estimate
The
estimate is based on the
residual sum of squares:
a;.
A
a;
A
SS sis the value of S in Eq. (9-2) when the values f3o and {3 1 are chosen for the intercept
and slope, respectively. The expected value of the residual sum of squares is
E(SSJ
=(/- 2)ae2
2.
.
d
so th at ae 1s estimate as
(9-6)
SSs is the estimate of the error variance and MS 6 is called the mean square error. The
square root of MS s is the standard error of regression. SSs has (/- 2) degrees of freedom
because two degrees of freedom are associated with the regression estimates of the slope
and intercept.
Statistics for Petroleum Engineers and Geoscientists
215
Returning now to the sample slope and intercept variabilities, our estimates are based
on Eqs. (9-5) and (9-6).
(9-7a)
and
2
MSs
(9-7b)
s{31 = I
2: (Xi-
X)2
i=l
The variabilities of both sample parameters are directly proportional to the variability
of e and inversely related to the variability in X. Thus, a greater spread in X's will give
more reliable parameter estimates. This means that a well-designed experimental program
will interrogate the explanatory variables (the X's in this case) over as wide a range as
possible.
Example 1 c- Estimating Parameter Variability. We now estimate the
variabilities of the sample intercept and slope for the small data set of
Example la. The residuals are shown in the table below.
i
1
2
3
4
X·!
y.
0
1
0
1
3
4
2
1\
I
1
1\
Yi
Yi- Yi
(Yi- Yi)2
0
2
2
4
0
-1
1
0
0
1
1
0
-
xi- X
(Xi- X)2
-1
0
0
1
Hence, sst:= L(Yi- Yi)2 = 2, MSt: = SSs/(I- 2) = 1, X= 1 and
'L(Xi - X)2 = 2. This gives
2
1
12
s = 1(- + -) = 3/4
f3o
4 2
and
2
s{31 = l/2
If we assume that the errors are normally distributed, we can produce confi-
dence intervals for '/Jo and '/3 1 from the above results (Chap. 5). The 95%
interval for '/30 is± t(0.025, 2) s130 = ±4.3 (0.75) 112 = ±3.7. The t value
1
0
0
1
Chap. 9 Bivariate Analysis: linear Regression
216
chosen, t =4.3, reflects the two-sided nature of the interval and that, with 4
data and 2 parameters estimated, there are only two degrees of freedom
remaining. A similar treatment for '/3 1 yields the 95% interval
±t(0.025, 2) sp1= ±4.3 (0.5) 112 =±3.0. From these data and the assumptions mentioned, f3o = 0 ± 3.7 and [3 1 = 2 ± 3 with a 95% level of confidence.
Such large errors on the slope and intercept reflect the small number of data in
this example.
If the errors remaining from the regression are independent and normally distributed,
then the estimates of the slope and intercept are called BLUE. This stands for best
(minimum variance), linear, unbiased estimators. We will encounter BLUE estimators
again in Chap.l2.
9-5
SEPARATElY TESTING THE PRECISION OF THE
SLOPE AND INTERCEPT
P1
Suppose we wish to test whether the slope
is significantly different from some a priori
2
[3 12 If the errors are N(O, a e) and the observations Yi are uncorrelated,
/31- N(/3 1, s13 ). To test whether the estimate and the constant are different, we form the
1
statlstlc
~onstant,
0
I
0
The t fJ statistic has a t distribution with (/ - 2) degrees of freedom (Chap. 7). The t
distributioh is used here to account for the added variability in the test resulting from the
M S t: approximation of s2 for small data sets. As I becomes large, tfJ approaches an
N(O,l) distribution. The regression estimates account for two degrles of freedom;
therefore, the t ratio has df=l- 2. We complete the test by comparing the computed
value of tf3 with t( a/2, df). The proposed value and the estimated value are different to
the a probJbility if t p1 is greater than the tabulated value.
p
A similar approach can be taken for the confidence limits of 0 . To compare
any reference value [3~. we use
Po with
Statistics for Petroleum Engineers and Geoscientists
Example 2a -Separately Testing Intercept and Slope Estimates. A set of
core-plug (C/Jp) and wireline (C/Jw) porosity data (I= 41) were depth-matched and
plotted to assess their relationship. The aim of the analysis is to see if C/Jw
could be used to predict fPp· If both measurements are responding to the same
features of the formation, we would expect a regression line to have both unit
slope and zero intercept. The result (Fig. 9-4), however, suggests the
measurements may not be similar; the regression-line intercept is not zero and
the slope is 16% greater than one. The slope and intercept estimate
variabilities should be evaluated to help decide what these data are indicating.
We will test the intercept and slope to see if either one is statistically different
from 0 or 1, respectively.
0.20
$
.=ell
~
0.16
~
~
8
0.14
0.12 - - 1 - - - - - v - - - - . , . . - - - - - ,
0.12
0.14
0.16
0.18
Wireline <Pw
Figure 9-4. Wireline and core-plug porosities with regression line.
We first check the residuals, fPpi- $pi· A probability plot (Fig. 9-5 top left)
suggests that they appear normally distributed. A plot of residuals against C/Jw
(Fig. 9-5 top right) shows no particular pattern, suggesting the linear model
is adequate. Similarly, there appears to be no pattern in a scatter plot of the
217
Chap. 9 Bivariate Analysis: linear Regression
218
ith residual versus the (i+ l)th residual (Fig. 9-5 bottom), suggesting the
errors are uncorrelated to each other. These checks help to confirm that the
assumptions of e- N(O,a~ and the appropriateness of the linear model
apply.
For these 41 data, MS.e = 0.0000892, C/>w
0.00342. From Eqs. (9-7),
2
1
s130 = 0.0000892 ( 41
= 0.159, and L(</Jwi-
- 2
</Jw) =
0.1592
+ 0 _00342) = 0.00066
and
2 = 0.0000892 = 0 0261
0.00342
·
s/31
The critical t value for the 95% level is t(0.025, 39)
-0.016 - 2.02-! 0.00066 <
f3o
-0.068 <
f3o
=2.02, so that
< -0.016 + 2.02-l 0.00066
< 0.036
and
1.163 - 2.02-10.0261 ~ [31 < 1.163 + 2.02-l 0.0261
0.83 <
lh
< 1.5
Hence, the data do not contradict either of the hypotheses that
f3o
= 0 or
[3 1 = 1. The t tests confirm this:
1-0.016 - 01
--Jo.ooo66
0.62
and
t = 11.163- 11 = 0.99
[Jl
0.0261
-v
Referring to a table oft values (e.g., Abramowitz and Stegun, 1965), we find
that t8 is only significant at approximately the a= 50% level, and tp is
signifi~ant at the 70% level. This analysis suggests that the wireline 1
measurements can be used interchangeably with core-plug porosities provided
the formation character does not change from that represented by these data
(e.g., similar levels of porosity heterogeneity).
Statistics for Petroleum Engineers and Geoscientists
2.
2.0-
<:;)
c::::>
1.0-
...·
oil
~
tpEi
'-"'
00
-....
/
0.0-
00
-2.0I
-3
..
..
1.
0.
..
..
...
....
..
..."'"'
-1.0-
<lJ
~
"'""
.l
~
::I
"e
219
I
-2 -1
I
I
I
0
1
2
-2.0
0.14
3
0.15
~
·;s.,
"'+
"'"
·e.
-e-
-
...
..
0.010.00-.
..
'-"
~
::::1
...."0
0.18
0.02
+
<e-
0.17
<!>w
Normal Quantiles
-
0.16
-0.01-
<;f.!
<I)
~
-0.02
-0.02
I
I
I
-0.01
0.00
0.01
Residual ( <j> pi•
0.02
/1.
-
<!>
pi )
Figure 9-5. Diagnostic probability (top left) and residuals plots (top right
and bottom) suggest the errors are normally distributed, show no
systematic variation over the range of X, and are uncorrelated.
Chap. 9 Bivariate Analysis: Linear Regression
220
As an alternative to plotting the residuals as in Example 2a, we could have estimated
the co variances between the residuals and lf>w and between the ith and (i+ 1)th residuals to
perform these functions. Plotting is actually a more powerful way to check for
independence because it will show all error types.
9~6
JOINTLY TESTING THE PRECISION OF THE
SLOPE AND INTERCEPT
In Example 2a, we separately tested the slope and intercept estimates and concluded that,
at a 95% level, the sample line does not have a slope or an intercept appreciably different
A
from what we would expect. f3o was tested with theA null hypothesis (Ho) that f3o = 0 and
the alternative hypothesis (HA) that f3o 7: 0. For /3 1, H o is /3 1 = 1 and H A is /3 1 7: 1.
What we did not do, however, was test together (jointly) the null hypothesis that f3o =0
and f3 1 = 1. In Chap. 7, we explained that applying a t test to each part of a joint
problem does not give the same confidence level that a joint test does. In situations
where the estimates are independent, the confidence level decreases (e.g., a increases from
5% to 10% for two independent estimates tested jointly). In this case, the result is more
l\
A
complicated because f3o and /3 1 are correlated.
The problem of joint confidence tests can be described in geometrical terms. Separate
confidence intervals (C. I.) on '/30 and 1 provide a rectangularly shaped confidence region
in the /30 - /3 1 plane (Fig. 9-6 left). This region represents the values for /3 0 and /3 1 that
the data would support within the specified confidence level. The value for one parameter
takes no account of the effect it has on the other parameter. The borders of the region
represent different values of S in Eq. (9-2), where S is the sum of squared deviations in Y
between the regression line and the data.
fi
Regions of constant S are ellipses (Fig. 9-6 right), and it is values of S that are a
better measure for permissible combinations of f3o and /3 1. Why is this so? It is clear
l\
l\
that S is a measure of the joint suitability of f3o and [3 1 with the data (Eq. (9-2)), butS is
also a random variable. Therefore, S has an associated confidence interval so that S::;; Sa,
where Sa is the maximum permitted value of Sat the (1 - a) level of confidence. S may
A
A
not achieve its minimum value when f3o = f3o and /h = {31. The data and the least-squares
estimator, however, dictate the values for '/30 and 1. Therefore, this confidence level for
S represents all pairs (/30 , {31) that make the joint null hypothesis acceptable such that
S ::;; Sa· For example, points A and B are both on the border of the independent
assessments region (Fig. 9-6 left) but have different values of S. Point A is well outside
fi
Statistics for Petroleum Engineers and Geoscientists
221
the permissible variation of S at the specified confidence level whereas Point B just
qualifies.
The equation for the elliptical region, derived in Montgomery and Peck (1982,
pp. 389-391), is
where F(2, I - 2, a) is the F statistic (see Chap. 7) for the desired confidence level
(1 - a).
'---1--"----t--pl
A
pl
Figure 9-6. Confidence regions for f3o and /31, as reflected by the data, for
independent assessments (left) and joint assessment (right) are
shown by the shaded regions. (C. I. =confidence interval.)
Example 2b- Jointly Testing Intercept and Slope Estimates. We now test
the slope and intercept estimates jointly at a 95% level of confidence with
Ho: f3o = 0 and f31 = 1. We use Eq. (9-8) with the following values:
~0 - {30 = -0.016 - o = -0.016;
I= 41
"Lxf = L(X i-
MSe
/11 - {31 =1.163 - 1 = 0.163
= o.oooo892; "Lxi =I
x)2 + I
x = 41•0.159 = 6.5
X2 = o.00342 + 41·0.159 2 = l.04F(2,
3.23
39, o.05) =
222
Chap. 9 Bivariate Analysis: Linear Regression
The left side of Eq. (9-8) gives
41(-0.016)2 + 2(-0.016)(0.163)6.5 + (0.163)2 1.04 = 22
2(0.0000892)
which exceeds the critical F value of 3.23 by a considerable margin. Hence,
the data do not support the model Y =X (i.e., a model with unit slope and
zero intercept).
The contours of constantS (confidence regions) are rather long, narrow, and
1\
1\
inclined with a negative slope for this data set (Fig. 9-7) because Po and p1
1\
1\
are strongly anticorrelated: if p1 increases, Po will decrease. From
1\
1\
Eq. (9-5a), the variability of Po is caused by the variabilities in Y and [3 1 .
For this data set, where Xis far from X= 0 (about two standard deviations),
1\
the variability of fh dominates. In the context of Example 2, we can
conclude that there is not a one-to-one correspondence between core-plug and
wireline porosities, contrary to the independent assessment results, because
this test overlooked the dependency between slope and intercept.
0.06
Independent assessment
0.03
A
0.0
~o-
-
-
-
-
-
-0.03
-0.06
-0.09
~1
+--.----.--....-~~+---t-.---T~"'--..::,.--..,---=
0.2
0.4
0.6
0.8
1.0
1.4
1.6
1.8
Figure 9-7. The rectangular 1\confidence
region obtained from independent
1\
assessments/\of [3 0 ~nd [3 1 and the joint, elliptical confidence
region of Po and P1·
223
Statistics for Petroleum Engineers and Geoscientists
There are other methods for producing joint confidence regions and testing besides the
constant-S approach discussed here (e.g., Montgomery and Peck, 1982, pp. 393-396).
They are generally simpler but may not give the desired result if a specific confidence
level is required; they only assure that the test will be at least at the specified level. Joint
confidence regions are also useful for testing the significance of different regression lines
(Seber, 1977, Chap. 7).
9e 7
CONFIDENCE INTERVAlS FOR THE MEAN
RESPONSE OF THE liNEAR MODEl AND
NEW OBSERVATIONS
If we wish to estimate E(Y I X= X0), an unbiased point estimator is
A
Y 0 has variance
A
A
1\
Var(Yo) = Var (/3o + /hXo)
-
A
+
:L
-
= Var[ Y + f3I(X 0 - X)]
=
When we replace
a;
2 [1
ae I
(X 0 - X)2
(Xi -
J
X)2
(9-9)
by its estimate, this gives the following sample variance:
A
The estimate for Var(Yo) is then used to define the (1 - a) confidence bands for the
estimate of E(Y I X= Xo):
A
Y 0 ± t( a/2, I - 2) .\})
0
Chap. 9 Bivariate Analysis: linear Regression
224
The confidence interval widens as (X o - X)2 increases and, when
1\
x0
= 0,
1\
Var(Yo) = Var(/30). Intuitively, this result seems reasonal:>le since we would expect to
estimate E(Y I X= Xo) better for X values near the center of the data than for X values
near the extremes.
To produce a confidence interval about the line to define where a new observation
(X0, Y*) might lie, we add the error component variability to Eq. (9-9).
.
1\
2[
1
Var(Y*) = Var(Yo) + Var(e) = a-e 1 + I +
.
(X 0 -
X) 2
J
L (X. _ X) 2
I
Addition of variances is permissible here because of the independence of the error e.
Using sample values, this leads to the confidence interval
1\
Yo± t(a/2,1- 2)
about the line for a new observation, Y*.
9m8
RESIDUALS
ANAlYSIS
As suggested in Example 2, plots of the residuals can be very helpful in determining how
well a model captures the behavior of the data. Residual plots can also convey whether
parameters that were not included in the regression could help to better predict the Y's.
Example 3 -Residual Analysis of a Data Set. From the scatter plot in
Fig. 9-8 left, a linear relation looks appropriate and a least-squares line may
be calculated. A residuals plot (Fig. 9-8 right), however, shows a problem.
For -1 <X< 4, the residuals systematically increase with X, suggesting the
calculated line is not increasing as quickly as the Y's are. The point at
X= 5.8 (a high leverage point) has a large influence on the line and has
reduced the slope. Consequently, there is a significant relationship between
the residuals and X. We should now decide whether to keep the point, which
requires further assessment of how representative it is. It would be
inappropriate to continue with the line as it is because the assumption of
normally distributed, independent errors has been violated.
Statistics for Petroleum Engineers and Geoscientists
8.0
225
1.0
6.0
-=
0.5
0
10
[I)
;;... 4.0
'0
';;)
2.0
0.0
-2.0
0.0
2.0
X
4.0
6.0
oO
0.0
0
-0.5
-1.0
-2.0
~
0
0
0
101
~
0
0
~
0
0
0
0
0
0
0.0
2.0
4.0
6.0
X
Figure 9-8. While a linear model (left) appears suitable, a residual plot
suggests otherwise. Note the change of scale for the vertical
axis between the plots.
As already suggested, we can plot residuals against any other variable, including those
not considered in the model. By doing so, we will be able to tell whether there is any
variation in Y that can be explained by a missing variable. This approach is an
improvement over multiple regression procedures that throw all possible variables in at
once without considering whether they have any explanatory power. We should approach
multiple regression systematically, variable by variable, choosing first those in which we
have the most confidence and those that have the strongest engineering and geological
reasons for being included. Further aspects of residual analysis are discussed in many
regression texts, including Box et al. (1978), Montgomery and Peck (1982), and Hoaglin
et al. (1983).
Residual analysis is particularly important when confidence intervals are desired.
Confidence intervals can be used, when the assumptions are fulfilled, for many purposes.
They help to decide whether a predictor has any explanatory power, to determine required
numbers of samples, and to compare parameter estimates with other values. Confidence
intervals, however, may easily mislead if the residuals are not examined first to ensure the
model and errors are behaving suitably.
A simple example of misleading results occurs when confidence intervals are used to
decide whether X has any explanatory power for Y. Figure 9-9 shows two scatter plots
and fitted lines for which the slope confidence intervals include zero. This result for the
left figure correctly leads to the conclusion that X has no power to predict Y. On the
other hand, the right figure suggests that X has considerable explanatory power, but the
Chap. 9 Bivariate Analysis: Linear Regression
226
A
model is inappropriate. In both cases, a plot of (Yi - Yi) versus X would immediately
show whether the fitted line properly reflected the X-Y association depicted by the data.
y
y
4.0
1.0
Y = -0.043X + 1.809
0
0
0.8
0
(I
0
0
0.6
0
0
0
0
0
0.2
0
0
0
0
(I
0
0
0
0
1.0
0
00
0
oo o
0
0
0
0
Y =O.OOlX + 0.495
0.0
(I
0
0
2.0
0.4
0
3.0
0
0
0.0
0
9-9
5
10
0
5
X
X
Figure 9-9. Two data sets with near-zero regression line slopes but
different explanatory powers of X.
10
THE COEFFICIENT OF DETERMINATION
Often we would like to know how much of the observed variability can be explained by a
given regression model with fitted parameters. The coefficient of determination is that
proportion of the variability in Y explained by the model. It is defined as
R2 = 1- SSe
SSy
where SSy is the total variability in Y, given by I, (Yi - ¥)2, so that
R2
= 1 _ 2& i
- Y i) 2
L(Yi- Y) 2
The total variability in Y, SS y, can be broken down into two components: the part
that the model can explain, when X is known and can be used to predict Y, and the
remainder (residual) that the model cannot explain. R2, so-called because of its
relationship to r 2 when X and Yare joint normally distributed (Chap. 8), measures the
residual part compared to the total variation. If R2 = 1, there is no residual and the model
Statistics for Petroleum Engineers and Geoscientists
227
explains all variation in Y. When R2 =0, the residual and total variability are equal, thus
the model has no explanatory power.
With a little reflection, we see that the coefficient of determination
1. increases as the inclination of the cloud of points increases and vice-versa,
2. does not measure the magnitude of the slope of the regression line, and
3. does not measure the appropriateness of a model.
The first point suggests that, given a cloud of points and a regression line with a positive
slope, R2 will increase as the cloud is rotated counter-clockwise about the point
( X, Y). Thus, while the appropriateness of the line to the cloud does not change, R2
changes. The second point follows because the definition of the coefficient does not
contain the slope. The third point is a form of the oft-repeated caveat about regression
not being able to determine the correct model.
Caution should be used when comparing R2 values. We have just observed that
comparisons of identical models between different data sets may be invalid using R2. But
comparisons of different models using the same data set may also be misleading. For
example, SU,QQOSe we wish to determine whether the model Y = f3o + f31X is better than
the model "'-/ Y = /32 + f33X for a particular data set. If we simply compute the values of
R2 for the two cases and compare them, we will be comparing two different things. In
the first case, R2 gives us the proportion of variability in Y that is accounted for when X
is known. In the second case, it is the proportion of variability in Tithat is accounted
for by knowing X. Because the square root is a nonlinear function, the variability in Y is
quite different from the variability in fY (e.g., compare Y decreasing from 100 to 0 with
what happens to {f). To compare the two models, we should compare R2 for the model
Y = (f32+f33X)2, where /32 and /33 are estimated by regressing ..fY upon X, with the R2
value obtained using the model Y = f3o+f31X.
Example 4 - R2 and Porosity-Permeability Relationships. Example 6 of
Chap. 3 discussed the lower Brent sequence, showing that geological
information can help to separate porosity and permeability data from
geologically different elements. This is also true for porosity-permeability
relationships. Figure 9-10 left shows a typical plot and regression line that
would be obtained if all data from the Rannoch formation were included. A
large R2 value obtains for the regression line because the line connects two
clouds of data, well·separated in terms of their porosities. The lower-left
cloud represents carbonate-cemented regions (concretions) with very little
porosity and permeability. The upper-right cloud represents micaceous, fine-
Chap. 9 Bivariate Analysis: Linear Regression
228
grained sandstones with no permeability below 7 mD. The regression line
connects these two groups as if they represented a continuum of changing
porosity and permeability, yet they represent very different lithologies with
no permeability between 0.03 and 7 mD. The line poorly represents the
sandstone porosity-permeability relationship (e.g., it underpredicts
permeability for porosities below 25%) and is not needed to predict the
concretion properties. The concretions are nonproductive intervals, easily
detected with resistivity or acoustic measurements. A more appropriate
regression line for the sandstones is obtained ignoring the concretion data
(Fig. 9-10 right) but, for this line, the R2 has decreased considerably.
Q
1000
1000
100
100
10
10
E
~
:s
E
...
.1
~
.01
.1
.01
.001
.001
0
10
20
Core Porosity, %
Figure 9-10.
30
40
0
10
20
30
Core Porosity,%
Rannach formation porosity-permeability data and
regression lines including (left) and excluding (right) the
carbonate concretions.
9-10 SIGNIFICANCE TESTS USING THE ANALYSIS OF
VARIANCE
The breakdown of the total variability of Y into two parts can be used to develop a
significance test for the model. The total variation in Y can be expressed as
SSy = SSreg +SSe, where SSreg is the regression sum of squares, L (Yi- ¥)2.
Following the discussion in Chap. 7, the ratio of the regression to the error sums of
squares forms a statistic suitable for the F test.
40
Statistics for Petroleum Engineers and Geoscientists
F-
229
SSreg
SS~/(/-
2)
can be tested against F(l, I- 2, a) at the desired confidence level, 1 - a, with the null
hypothesis Ho: fJ1 = 0 and the alternative !!A:_f3r! 0. SSreg has only one degree of
freedom since the line must pass through ( X, Y).
Example 5- Interpreting Computer Package Regression Output. In
Example 1, a regression line was obtained for the data set {(X, Y)) = { (0, 0),
(1, 1), (1, 3), (2, 4)}. A typical computer printout of this process gives an
/\
/\
analysis of the slope and intercept estimates, {3 1 and {3 0 , along with their
standard errors. The t ratios and probabilities commonly assume the null
/\
/\
hypotheses are {3 0 = {3 1 = 0, and care is needed to remember that these null
values might not suit the user. For example, we may be more interested in
/\
whether /31
= 1.
Y by X
Linear Fit
Summary of Fit
Ftsquare
0.8
Mean of Response
2
Root Mean Square Error
Observations
Analysis of Variance
DF Sum of Sguares
Source
1
8.000000
Model
2
2.000000
Error
C Total
3 10.000000
Mean Sguare
8.00000
1.00000
1
4
F Ratio
8.0000
Prob>F
0.1056
Parameter Estimate
Thim.
Intercept
X
Estimate
0
2
StdError
0.86603
0.70711
t Ratio
0.00
2.83
Prob>ltl
1.0000
0.1056
The analysis of variance F test gives the same result as the t test on the
slope. The model sum of squares is SSreg and the error sum of squares is
SSE. The mean square values are sums of squares divided by their respective
degrees of freedom.
Chap. 9 Bivariate Analysis: Linear Regression
230
9m11 THE NOaiNTERCEPT REGRESSION MODEL
A frequently encountered special case of the linear regression model is where the intercept
is zero. The number of degrees offreedom for the model is now (J- 1) since we have fixed
the intercept, but all of the other equations are the same. The model is
The least-squares estimate of the slope is
L(YiXi)
/1.
fh
=
2
LX·l
which is unbiased as before. The estimate for
(J'~ is
1
/\
MS £ =-~(Y·-Y·)2
/-lL, I
l
The confidence interval for the slope is
/\ ± t(a/2, Ifh
1)
~·s£
-2
LX·I
The confidence interval for the mean response at X.= Xo (which becomes wider as x 0
moves away from the origin) is
'
Y 0 ± t( a/2, I - 1)
For the no-intercept model, the coefficient of determination is given by
Statistics for Petroleum Engineers and Geoscientists
231
The denominator of this expression is a sum of squares about the origin, not the mean
of Y. Hence, comparisons between coefficients of determination from a no-intercept
model and a with-intercept model are not meaningful.
All the above variabilities also apply for the more general case where the line is forced
through an arbitrary point (X0, Y0), not just the origin. In this case,
ftl =:EX iyi- ~o:EY i 2- Yo'LX i + IXoYo
/\
and
/\
f3o = Yo - f31Xo
'LXi +IX 0 - 2X 0 :Exi
9-12 OTHER
CONSIDERATIONS
As we noted in the introduction to this chapter, least-squares regression is mathematically
quite straightforward but it has numerous statistical ramifications. We will cover some of
the more advanced aspects in Chap. 10, but there are basic topics we still have not yet
discussed. We now consider some of these issues.
Model Validity
Strictly speaking, regressed models may not be valid outside the range of the variables
used to fit the model. Of course, if the model to be regressed is based on physical
principles, this is much less of a factor, but regression-only models are often not so
based.
The risk of extrapolation is especially large when a regression model is developed from
data in one portion of a reservoir and then applied blindly throughout the reservoir. The
computer predictions may be based upon values of X well outside the range for which the
original relation was developed. The geological properties of the rock can also change,
invalidating the X-Y model, as was shown in Example 4. The fundamental problem is
that we often employ bivariate models to multivariate problems when we cannot measure
the other explanatory variables. Some of these hidden variables are geological factors
such as grain size and sorting, minerals, and diagenetic alteration that, in nearby wells,
may not change significantly. Over larger distances, however, these variables cause the
reservoir petrophysical properties to change considerably, invalidating predictions that do
not take these factors into account.
232
Chap. 9 Bivariate Analysis: Linear Regression
Situations With Errors in the Explanatory Variable(s)
We have only considered those situations where the X values can be perfectly measured;
all the error was associated withY or the model (e.g., unmeasured variables). In practice,
this means that the error in X should be much less than the errors in Y and in the model
for these techniques to apply. When substantial errors exist in X, more involved methods
are available that either require that the ratio of the errors in Y to those in X be known or
that X no longer be an arbitrary value. If conventional procedures such as ordinary least
squares described above are used, the relation will not appear as strong as it should be,
A
causing /31 to be too small (negative bias). Seber (1977, pp. 155-160) has further details.
In any case, if X cannot be measured with reasonable accuracy, its value as a predictor is
diminished and it may not be worthwhile to develop a predictive model including that
variable.
Predicting X From Y
Although we have only considered the case of using least-squares regression with a model
to predict E(Y) on the basis of X, the alternative situation (predicting E(X) on the basis of
Y) may also occur. The lines are different for these two cases. Although the line will
still pass through ( X, Y), the slope will change from SSxy/S~x to SSxy/SSy. This
is because the minimization is now on the X residuals, Xi -Xi· Compare this to
Fig. 9-2 and Eq. (9-2). In this case, Y is assumed to be knowR with neg~igiblK error.
Algebraic manipulation of the Y-on-X model to predict X (i.e., Xi = (Y i - f3o)l fh) will
give biased results except at ( X, Y) and is not recommended One exception to this rule
will be discussed in Chap. 10 concerning prediction of the X-axis intercept.
If the Y's are measured for prespecified values of X, then the least-squares regression
procedure to predict X from Y does not apply. The procedure assumes that the response
variable is a random variable. For example, suppose we have a core flood for which, for a
given amount of injected fluid (X pore volumes), we measure the oil recovery, Y.
Developing a regression model to predict Y from a knowledge of X is appropriate,
whereas applying regression to develop a predictor of X from Y violates the procedure and
gives a meaningless result.
It may seem curious that more than one line is available to express relationships
between two variables. In deterministic problems, one and only one solution exists; in
statistical problems, each line represents a different aspect of the relationship. Y-on-X
regression estimates E(Y I X= X0), X-on-Y regression gives E(X I Y = Y0 , while other
lines express yet other aspects of the X-Y relationship. This variation between different
lines is in addition to the variability that exists because we have a limited sample of X-Y
values.
Statistics for Petroleum Engineers and Geoscientists
233
Discrete Variables
While we have considered only continuous explanatory variables, it is also possible to
use discrete variables. For example, we may wish to develop a drilling model relating the
penetration rate (Y) to bit type (X). Clearly, bits come only in distinct types, so we have
to consider discrete variables of the form X= 1 (roller cone) or X= 0 (synthetic diamond).
If J bit types are involved, (J- 1) regressors will be required.
Variable Additivity
Once again, during both the model development and parameter estimation phases, the role
of additivity of the variables should be considered. Chapter 5 discussed the considerations
and these apply in regressioh as well. The problems are greater with regression, however,
1\
~ecause of the presence of cro~s-product terms (J:xiYi) and the fact that estimates f3o and
/31 have standard errors.
· When a physically based model exists for the X-Y relationship, additivity questions are
usually less problematic. The model parameters and cross-product terms may have a
physical interpretation. For example, porosity 1/J and bulk density Pb are related by mass
balance through Pb =Pma + rfJ(p1 - Pma) for a given volume of material, VT• where Pma is
the matrix density and p1 is the pore-fluid density. A regression of Pb upon 1/J for samples
of equal bulk volume provides estimates of Pma and p1 - Pma· Standard errors of these
estimates are masses, since multiplication by Vr is implied, and so are additive. The
cross-product term rfJPb also represents mass. .
Without a physically based model, regression introduces questionable operations. For
example, a permeability predictor in the form log(k) using porosity is often sought.
Regressing log(k) upon r/J involves calculations with terms f/llog(k) for which a physical
interpretation is difficult.
9-13 SUMMARY REMARKS
Bivariate regression is an elegant method for eliciting a predictive relationship from data
and a model. The results, however, are only as good as the care taken to ensure the model
is appropriate and the underlying assumptions are tolerably obeyed. Diagnostic
procedures are readily available for ensuring the model and data are consistent and for
234
Chap. 9 Bivariate Analysis: Linear Regression
examining for outliers. In this regard, the residuals are extremely valuable and will show
model adequacy as well as the value of including further explanatory variables in the
model.
Estimates of the model parameters can be assessed for variability, either separately or
jointly. These variabilities, based on the residual variability, give a valuable indication of
the explanatory power of the predictor variable. R2 also uses the residual variability to
indicate what proportion of the total variability in Y can be explained by a knowledge of
X. Inappropriate comparisons of R2 are easily made, however, because it is
dimensionless and the scale of variability is not apparent.
10
ALYSI
A
11!1
Ill
Having discussed the mathematical basis and statistical interpretations for the ordinary
least-squares (OLS) estimator in Chap. 9, we now explore more advanced topics of
regression. By and large, these topics do not require further mathematical methods or
sophisticated statistical analyses. They involve commonly encountered issues that require
a more careful consideration of the assumptions involved in OLS and their implications.
We will try here to highlight the statistical aspects of the problem for which the OLS
method provides estimates.
1 0=1 AlTERNATIVES TO THE LEAST-SQUARES UNES
The least-squares procedure, described in Chap. 9, is remarkably robust even if there are
deviations from the basic assumptions (Miller, 1986, Chap. 5). The most .Q!Ominent
weakness of the procedure is for possibly erroneous data with large IX - XI. These
extreme data points have a large leverage (Chap. 9, Example 3). Robust methods exist to
develop regression lines, but variabilities in slope and intercept are more difficult to
obtain. Hoaglin et al. (1983, Chap. 5) is a useful introduction.
Other lines, such as the major axis and reduced major axis lines (RMA) also exist but,
as shown in Chap. 8, they are not regression lines. One attraction of the RMA line is
that it lies between the Y-on-X and X-on-Y lines (Fig. 8-6 and Eqs. (8-3) and (8-4)). The
RMA line slope (ay/ax) is between the Y-on-X line slope (pay/ax) and X-on-Y line
235
236
. Chap. 10 Bivariate Analysis: Further Linear Regression
slope (p-luy/ux) and they all pass through ( X, Y). It appears that the RMA line
might provide better estimates for Y than the OLS line for the case where X, the
predictor, is measured with significant error. Since the Y-on~x slope will underpredict
E(Y I X= Xo) and the X-on-Y slope will overpredict this value, a line with an
intermediate slope (the RMA line) might be better. This iogic, however, has two
weaknesses.
The first weakness concerns the assessment of the relative errors of measuring X and
Y. If the error term e in the model Y = f3o + /31 X+ e represents only measurement error
of Y, then substantial errors· in X would affect the estimates
and
obtained
from Y-on-X regression. The error term e, however, often includes model inadequacy.
· Hence, the errors of X should be compared to the errors arising from model inadequacy as
bo
b1
well as ¥-measurement errors.
The second weakness concerns the utility of developing a model that is to be used with
unreliable predictor values. When X is poorly measured, the Y-on-X slope will approach
zero, indicating that, under the circumstances, the best estimate of Y is~= E(Y). This
is a useful diagnostic feature indicating that, no matter how good the underlying X-Y
relationship is, the measurement errors of X render it nearly useless as a predictor of Y.
The relative merits of lines should be assessed in the context of the specific problem.
As a general rule, remember that, while the RMA Une portrays the X-Y covariation
without regard to the roles of the variables, the regression lines X-on-Y and Y~on-X do
assign explicit roles to the variables. Thus, use of the RMA line in a prediction role
should be carefully justified.
A frequently overlooked limitation of OLS lines is that they may lead to biased results
for some applications. Recall from Chap. 9 that the re&l:ession line is an estimate of the
conditional expectation E(/1 X = Xo) and the estimates Y are unbiased. Thus, any linear
combination of estimates Y is also unbiased but nonlinear combinations will be biased.
The following simplified example illustrates this.
Example 1 a - Applying Regression Line Estimates for Prediction of Effective
Permeability. Permeability can be difficult to measure in situ. Therefore, it
is common in reservoir characterization to predict permeability from an OLS
regression line and in-situ porosity measurements. The regression line is
based on permeability and porosity data taken from geologically similar
portions of the reservoir. These predictions are then often combined, using
the arithmetic and harmonic averages, to predict the larger-scale aggregate
horizontal and vertical permeabilities of stratified rocks for subsequent
applications. (Recall from Chap. 5 that, in a layered medium, the arithmetic
average of permeability is an appropriate estimator for layer-parallel flow
Statistics for Petroleum Engineers and Geoscientists
while the harmonic average estimates aggregate penneability for layertransverse flow.)
A portion of a hypothetical porosity-permeability regression line and the
"data" are shown in Fig. 10c1left. The line is passing midway between the
two points shown, as we would expect (on average). Looking more closely,
the data indicate k may be 1 or 100 with equal probability when cp = 0.1. At
cp = 0.1, however, the line provides an estimate~= 50.5, which is neither 1
nor 100. Nonetheless, 50.5 is a reasonable value since regression lines are
designed to predict E(k I cp = c/Jo) and the average of 1 and 100 is 50.5.
Suppose now we have a rock unit composed of four equally thick and equally
porous layers (Fig. 10-1 right) with porosity measurements but the penneabilities are unknown. Therefore, we must use the regression-derived estimated permeability for each layer to predict the aggregate horizontal and vertical permeabilities of the unit. We compare the estimated horizontal and vertical permeabilities with the true values given as the arithmetic mean E(k I cp
= 0.1) = 50.5 (horizontal) and the harmonic mean [E(k-11 cp = 0.1)]-1 =
1.98 (vertical).
For the aggregate horizontal permeability, we have
4
kA. =±2: ti = 50.5
i=l
This agrees exactly,with E(k I cp = 0.1) = 50.5. For the vertical permeability
of the unit, the harmonic average gives
which is a poor estimate of the harmonic mean. From the regression
estimates, the rock unit appears to have no permeability anisotropy (verticalto-horizontal permeability ratio equals one), while the unit in fact has a
vertical-to-horizontal penneability ratio of 0.04.
237
238
Chap. 10 Bivariate Analysis: Further Linear Regression
100
0
Q
s
--- .....------
;;;; 50
;<::
==
~
~
.,.'.,. ...
~
<1>4 = 0.1 k4 = 100
~ 0 4-----~----~----------~
0.0
0.2
0.1
Porosity,
<!>
Figure 10-1. Regression line and data (left) and layered formation (right).
In Example 1a, the regression line gave an unbiased estimate for the arithmetic average
because this average is a linear combination of the estimates. This is a result of the
Gauss-Markov Theorem (Graybill, 1961, pp. 116-117), which assures that all linear
combinations of the estimates flo and /11 are unbiased. The harmonic average is a
nonlinear combination of the estimates, and use of the Y-on-X line could give highly
misleading results. The following example shows an alternative procedure for use with
the harmonic average.
Example lb- A Regression Estimator for the Harmonic Average. A
regression line based on resistivity to flow, r =k- 1, and using the data of
Example 1a would estimate E(r I cp = 0.1) as the average of 1 and 100-1, or
A
0.505. Thus, for the four-layer model of Fig. 10-1 right, ri = 0.505. This
gives, for the harmonic average,
kn =
(i ~4 ~ )-1 = 1.98
i
z=l
which agrees with the h2rmonic mean.
When the harmonic average is needed, better results are obtained using a y-1-on-X
line. The line then provides estimates of y-1 for use in the harmonic average. This is
still not entirely satisfactory, because the inverse of the sum of estimates is required.
Statistics for Petroleum Engineers and Geoscientists
239
Nonetheless, the sum would be much less variable than any one estimate of y- 1, so it
would be less biased. A similar approach, using a log(Y)-on-X line, would be suitable
for geometric mean estimation.
In Chap. 9, we remarked that the possibility of two lines (Y-on-X and X-on-Y) is a
curious feature of regression. This may be especially so for those more accustomed to
deterministic methods. Here, we have another example of regression being able to
produce several lines. These lines arise because the application of the estimates
influences the choice of estimator. This choice must be made on geological and physical
reasons; the statistics do not help.
10-2
VARIABLE TRANSFORMATIONS
The linear model is a compact and useful way of expressing bivariate relationships. In
cases where a linear model is inappropriate, some transformation of the variables may
permit a linear model to be used if the application for the estimates is compatible with
the transforms. For example, while porosity cp and permeability k are usually observed to
be nonlinearly related, a logarithmic transform (to any base) applied to permeability may
linearize the relationship: log(k) = f3o + /31 cp + e, and predicting log(k) is appropriate
for estimating the geometric mean. A particular case of linear bivariate relationships
occurs when random variables are joint normally distributed.. If random variables are
transformed to become joint normally distributed, the linear model will automatically
apply (Chap. 8). There are, however, some precautions that should be observed when
selecting a transformation and applying the results of the regression model.
Variable transformations affect the way that the noise, e, enters into the model. For
example, the modelln(Y) =f3o + /31 X+ e assumes the noise is additive for ln(Y). This
means that the noise is multiplicative for Y because Y = expCf3o + {3 1X) • exp(e). This
model is often appropriate when the size of error is proportional to the magnitude of the
g._uantity measured, e.g., e =±O~lOY. Regressing ln(Y) upon X then gives the parameters
f3o and f}1. If, on the other hand, an additive noise model is more suitable, we can regress
Y upon exp(X). This implies the model Y = f3o + {31exp(X) + e. The proportional effect
of noise in permeability measurements may be why ln(k) versus porosity (cp) plots are
more common thank versus exp(cp) plots. Recall from Chap. 9, however, that the term
e includes all errors, including model deficiencies as well as measurement errors. Thus,
while & small component of the k error may be multiplicative from measurement error,
the predominant component may still be additive, justifying a regression of k upon
exp(cp).
Chap. 10 Bivariate Analysis: Further Linear Regression
240
What happens if we get the noise model wrong? Usually, the spread of residuals varies
with X (Fig. 10-2). In this case, the assumption of OLS that c,-N(O, ~) (see Chap. 9)
is being violated since the variance is not constant. Alternative transformations of X and
Yare needed to produce a more appropriate noise model.
0.8
Residuals
0
0
0.0
-0.8 -+-------r-------.-0.1
0.2
0.3
X
Figure 10-2.
Regression residuals showing an inappropriate noise model.
Transformations of the response variable also affect the least-squares criterion and the
resulting line. This feature is quite evident, for example, with a logarithmic
transf~rmation (Fig. 10-3). Deviations above the regression line (log 10 (Yi) log 10(Yi) > 0) do not represent the same magnitude as deviations below the line
1\
(log 10 (Yi)- log 10 (Yi) < 0) in the Y domain. This behavior is because of the
nonlinearity of the logarithmic function. The effect can produce a considerable bias,
especially with highly variable data having a weak X-log(Y) relationship, but, in certain
circumstances, can be compensated for during de transformation.
Detransformation is the process whereby estimates of W =f(Y) produced by the
1\
1\
regression line are converted back to estimates of Y. The algebraic approach Y=J-l(W)
does not always work well whenfis strongly nonlinear or the relationship is weak.
However, when the assumption of e -N(O,J;) applies for the model W = f3o + {31X + e,
we can use the theory of the normal distribution to derive a bias-corrected Y.
Statistics for Petroleum Engineers and Geoscientists
241
For an.y regression of W upon X, E(W I X= Xo) = f3o + f3 1Xo. In the common case
W = ln(Y), Y will be log-normally distributed and W- N(f3o + f3 1Xo,
Thus we
have, from the properties of the log-normal PDF (Chap. 4),
a;).
E(Y I X= Xo)
= exp(fJo + fJ1Xo + 0.5~
This result indicates that there is an additional multiplicative factor of exp(O.So;) to
determining the conditional mean of Y besides the factor exp(fJo + f3 1X0) we might
naively assume.
is approximated from the data by the meari squared error, MS£
(Eq. (9-6)).
cr;
4 .................................................................................... ..
X
Figure 10-3.
The regression-line procedure gives equal weight to
deviations of9,000and 900.
The preceding equations apply equally well using base-10 logarithms as the Napierian
type. We suggest that, to. avoid making mistakes by overlooking factors of 2.3 to
convert In to log 10 , the Napierian form be used. The following example illustrates the
procedure for data transformed using base-l 0 logarithms.
Example 2- Detransformation of a Log-Transformed Data Set. The data of
Fig. 10-4 left indicate that the log 10(Y)- X relationship is linear, with the
line explaining about 79% of the variability in logw(Y). The sample
Chap. 10 Bivariate Analysis: Further Linear Regression
242
variance of log10(Y) is 0.733 and, with the line, this is reduced to
0.7~3(1-R2) = 0.153. This gives a standard deviation of about 0.4 decades
(= 0.153) about the regression line. A naive detransformation suggests the
relation = 1015.4X-1.41 (Fig. 10-4 right). A probability plot indicates
that the residuals of log(Y) are approximately normally distributed so that the
log-normal correction applies. This gives an additional factor of exp(o.so;>
= exp(0.5•2.32•0.153) = 1.50, where the factor 2.3 2 converts the log 10
variability to Napierian units (In). The bias-corrected line relation is
= 1.5•1Q15.4X-1.41, also shown in Fig. 10-4.
Y
y
15000
4.0
0
0
3.0
Bias-corrected Y
E
.3 2.0
1.0
oo
00
o
0.0
0.10
CODCD
o 8#
o dl
0
Log(Y) = 15.41X- 1.41
R 2 =0.791 s 2 =0.733
0.20
0.30
o~~~~~~~-0.10
0.20
0.30
X
X
Figure 10-4.
Data set for Example 2 with log- and untransformed plots.
At first glance, it may be difficult to discern whether the bias-corrected line is better
than the naive predictor in Fig. 10-4 right. More careful inspection reveals that the latter
line is consistently near the bottom of the scatter for large X; the bias-corrected line
passes more nearly through the center of the large-X data. Nonetheless, the linear plot
shows a considerable variability of the Y values that is not readily apparent on the log(Y)
plot.
W = ln(Y) is one example of the more general case of power transformations,
W = yP, discussed in Chap. 4. If we assume e- N(O,~, Jensen and Lake (1985) give
a general expression for any value of p. If p > 0, the correction is additive rather than
multiplicative. For example, when p
= 1/2,
E(Y I X= x0 ) = E(W I X= x0) 2 + cl-;4
. (10-1)
243
Statistics for Petroleum Engineers and Geoscientists
Jensen and Lake (1985) suggest transforming both predictor and response variables prior
to regression if the predictor is a random variable. The objective is to find functions f(Y)
and g(X) that make the joint PDF approximately normal. As described in Chap. 8, joint
normal variables automatically have linear regression relationships, which helps to keep
the X- Y relationship as parsimonious as possible. The noise model for f(Y) upon g(X)
will be appropriate and detransformation will be simplified since the residuals are
normally distributed. It is usually sufficient to select f and g so that the marginal
histograms are approximately symmetrical. This is not strictly adequate, however,
because symmetry of the marginal distributions is necessary but not sufficient for joint
normality. The following example demonstrates this procedure. Further examples are
contained in Jensen and Lake (1985).
Example 3- Transformations for Joint Normality. Scatter plots of a data set
(Fig. 10-5) suggest that neither untransformed nor log-transformed versions of
the response are entirely suitable, despite the impressive values of R2. TheY
plot shows a concave-upward form while the log(Y) plot is concave
downwards.
800
3
Y = 6550X - 1280 R
600
;;...
R2=0.92
400
0
200
Y = llX- 0.23
0
0
0.18
1~------T-------r------
0.22
0.26
X
Figure 10-5.
0.30
0.18
0.22
0.26
X
Scatter plots and regression lines for Y and log(Y).
Histograms of Y and log(Y) (Fig. 10-6left and center) are significantly
skewed while the histogram of X (Fig. 10-6 right) is reasonably symmetrical.
Consequently, only Y need be transformed. Since the sample PDF of Y is
right-skewed while log(Y) is left-skewed, an exponent between 0 and 1 is
2.44
Chap. 10 Bivariate Analysis: Further Linear Regression
indicated. A square-root transform (Fig. 10-7left) for Y gives a nearly
symmetrical histogram.
30
....
20
g
u
10
<
0
200
400 600 800 0
y
1
2
3 0.19
log(Y)
Figure 10-6.
0.24
0.29
X
Histograms for the data in Fig. 10-5.
1Y
The scatter plot and regression line for
(Fig. 10-7 right) suggest a better
linear relationship with X. There is not an even variability about the line;
lower values of X appear to have about twice the variability that larger X
values have. This might be corrected with more careful selection of
transformations, but it is unlikely to produce much benefit. If estimates of
Y, instead of yl/2, are needed, the final predictor should be bias-corrected for
the detransformation. The large R2 value and moderate variability of yl/2,
however, show that the correction is quite small (Eq. (10-1)):
Y= (200X -31)2 + ~1.16)2 =(200X -31)2 + 0.34
giving a correction of less than 1%.
In summary, there are three important aspects to using transformed variables in
developing models through OLS regression. The first is that using transformed variables
may help to keep the model simple because a linear model is sufficient for variables with
symmetrical histograms. Second, transformations help us to honor the noise behavior
assumed for OLS regression. Third, detransforming predicted values requires some care,
because the algebraically correct detransformation of predictions (i.e., naive
detransformation) may give significantly biased estimates.
245
Statistics for Petroleum Engineers and Geoscientists
30
30
yl/2 = 200X- 31
25
20
"S
=
Q
u
R2=0.95
yl/2
20
15
10
10
5
0
5
10
15
20
25
30
0.18
0.22
0.30
X
yV2
Figure 10-7.
0.26
Histogram, scatter plot, and regression line for
1Y.
10-3 WEIGHTED LEAST SQUARES
For ordinary least squares, we assume that all the data used in the regression method are
equally reliable. This assumption appears in the sum of the squared differences
(Eq. (9-2)):
I
S(/3o,f31)=
L
(Yi- f3o- f31Xi) 2
i=1
because all the differences (Yi- f3o- f31Xi) are equally influential on S. We can assign a
weight, w i• to each term in the sum
I
S*(f3o, /31) =
L
w i(Y i - f3o - f31X i) 2
i=1
A
A
A
If we calculate estimates {3 0 and {3 1 that minimize S*, then the equations for the f3 's
become
A
R
PO=
~ w t·Y·{3A 1 £...i
~ w.X·
I
I I
£...i
~Wi
.
246
Chap. 10 Bivariate Analysis: Further Linear Regression
and
where all summations are from i= 1 to i=l.
The w/S can be obtained in several ways. One way is to use experience. If some data
are thought to be twice as reliable as others, then the w's for the more reliable data should
be twice thew's for the less reliable points. It does not matter what values are chosen for
thew's (as long as they are not all zero), just make some twice the value of the others.
Another way is if quantitative assessments of the relative reliabilities of the Yi are
available, based on theory, for example, then more careful estimates can be made for the
w's. If Yi has variability a?, then wi
=a?
A
A
Small differences among thew's usually do not influence the estimates f3o and 13 1 by
much, compared to the OLS values. The changes will depend, however, on Xi and the
A
residual (Yi- ~· Residual analysis from ~eighted least squares requires that variables be
weighted by w i. Hence, plot -.J w i (Y - Yi) versus ~ w i Xi or any other desired
variable to assess the model fit.
10-4 REDUCING THE NUMBER OF VARIABLES
In a multivariate model such as Y =f3oU + /3 1V, it is tempting to reduce the number of
predictors by dividing by one of the predictor variables. Hence, Y = /3 1V + f3oU could
become
By setting Y * = YIU and X * = VIU, we obtain y* = f3 *1 X * + f3 *0 . Thus, we have
changed a three-dimensional problem into a two-dimensional problem and can use simpler
equations to estimate the f3 's. While this approach is algebraically correct, it may not be
A
A*
A
.1'.*
statistically sound. That is, will [3 1 = f3 1 and will f3o = f3 0?
Unfortunately, the answer is probably not. If U is constant, then the above procedure
is correct. If, however, U varies significantly compared to V, then the estimated f3 's will
differ. The reason is because of a change in the role of the errors in the models. When
we use OLS regression to estimate f3o and /3 1 in Y = f3oU + /3 1V, we are implicitly
Statistics for Petroleum Engineers and Geoscientists
247
assuming that the model is Y = f3oU + {3 1V + e. Furthermore, we're also assuming that
e does not change with U, V, or Y. Dividing the original model through by U gives us
the model
Y* = {3 1 x* + f3o + e*
where e*==e/U, which is not the same model as
Y* = f3 *1 X * + f3 *o + e
When U is constant, the noise e* is independent of V and Y but, if U is variable, then e*
is variable and is related to x* and y*. Hence, because we're solving two different
1\
1\*
1\
A* .
equations, we cannot expect /31 = /3 1 and f3o = f3 0.
This situation often arises in reservoir-engineering material-balance calculations.
Tehrani (1985) shows that the volumetric-balance equation can take the form
Y = f3oU + {31V. He proceeds to show that the revised model (Y/U) = f3o + {3 1(V!U)
1\
1\
does not give the same f3o and {3 1 as the original model and large errors can occur. The
following example illustrates the problem for a gascap-drive reservoir and shows that, in
addition, confidence intervals are also affected.
Example 4 -Material-Balance Errors in Gascap-Drive Reservoirs. The
material-balance equation for gascap-drive reservoirs is (Dake, 1978,
. pp. 78-79)
(10-2)
where F is the underground withdrawal (reservoir barrels, RB), N is the initial
oil in place (stock tank barrels, STB), E 0 is the oil and dissolved gas
expansion factor (RB/STB), Egis the gascap expansion factor (RB/STB), and
m is the ratio of the initial gascap-hydrocarbon to the initial oil-hydrocarbon
volumes prior to production. F, E 0 , and Eg are known quantities based on
measurements of the produced oil and gas and their properties. m may be
estimated from seismic or other geological data but, depending upon
circumstances, the estimate may be rather poor.
If m is not known or poorly known, we have a two-unknown, nonlinear
problem because Eq. (10-2) has the unknowns, m andN, appearing as a
product in the second term. Thus, the best solution procedure requires
nonlinear regression, a procedure outside the scope of this book. See Bard
(1974) for a good discussion of nonlinearregression methods and analysis.
We can solve linearized forms ofEq. (10-2) that, as we will see, have some
Chap. 10 Bivariate Analysis: Further linear Regression
248
limitations. The linear approaches require treating the product mN as one
regression unknown. If we do this, Eq. (10-2) can be solved directly as a
multiple-linear-regression problem with two explanatory variables, Eg and
E 0 , or by reducing the number of variables using the form
(10-3)
Thus, an OLS F/E 0 -on-EgfE0 line would also appear to give an interceptN
and slope mN.
For the following data, taken from Dak:e (1978, p. 92), we calculate lines for
both Eqs. (10-2) and (10-3) to examine the differences.
w-6F
Eo
Et.!
w-6FfE 0
EK/EO
5.807
10.671
17.302
24.094.
31.898
41.130
0.01456
0.02870
0.04695
0.06773
0.09365
0.12070
0.07190
0.12942
0.20133
0.28761
0.37389
0.47456
398.8
371.8
368.5
355.7
340.6
340.8
4.938
4.509
4.288
4.246
3.992
3.932
Because the units ofF are 106 or million RB, the corresponding units of N
will be million STB. Using multiple-regression (discussed briefly later in
this chapter), the zero-intercept, two-variable model (Eq. (10-2)) gives (with
standard errors in brackets)
N
= 105 [±34] and mN = 59.6 [±8.4]
With bivariate regression, the free-intercept, single variable model (Eq. (10-3))
gives
N
= 109 [±23] and mN =58.8 [±5.4]
The initial oil figures differ by about 3%, which is well within the statistical
variability of either estimate. The precision of the second model is
substantially less than that of the first model. For either model, it is clear
that material balance can only estimate the oil in place to within, say, 30 or
40% based on such few data.
m can be estimated by taking the ratio mN/N (i.e., 59.6/105 = 0.57).
Standard errors of mare more difficult to calculate because the regression
Statistics for Petroleum Engineers and Geoscientists
249
coefficients are correlated. A simplistic approach is to ignore the correlation
and use the sensitivity analysis described in Chap. 5. Let Z = mN so that
..1Z ..1m ..1N
-=-+z m N
where the ..1's are perturbations in the appropriate quantity. We have
8.4 ..1m 34
--=-+59.6 m
105
giving Llm/m = -0.18. Since m =mN/N = 59.6/105 = 0.57, m = 0.57 [ =t=O.l].
Note the errors in m are in antiphase with the errors of N.
A nonlinear regression with Eq. (10-2) gives the results
N = 105 [±34] and m = 0.57 [ +D.26]
The multivariate linear (the first model, Eq. (10-2)) and the nonlinear
regression results are identical for N. The estimates of m are identical but the
simplified perturbation analysis has considerably understated the standard
error. The simpler the solution methods to solve this problem, the less
reliable have been the results.
Why are the standard errors so different for the two linear regression models,
Eqs. (10-2) and (10-3)? This is because, by dividing by a common factor (£ 0 ), the
correlation between the response and explanatory variables has been improved. A simple
example discussed by Hald (1952, p. 614) helps to explain the problem. Consider the
case where a response (Y) and an explanatory (X) variables are unrelated so that the true
regression-line slope is zero. If new response and explanatory variables are created by
dividing by a common factor, y* = Y/Z and x* = X/Z, where Z is not constant, then y*
and x* will appear to be correlated and the regression-line slope will no longer be zero.
See Kenney (1982) for further examples of this effect.
1 0=5 THE X-AXIS INTERCEPT AND ERROR BOUNDS
Chapter 9 presented an expression, Eq. (9-5a), for the variability of the Y-axis intercept,
[30 . A fairly common but more COI)!plicated situalion is to obtain a confidence interval
for the X -axis intercept estimate -{30 I 1. This estimate is biased but there are few
p
attractive alternatives. In the statistical literature, this problem is often known as the
calibration or discrimination problem. There are several expressions for confidence
250
Chap. 10 Bivariate Analysis: Further linear Regression
/1.
/1.
intervals for -f3o !/3 1. Miller (1981, pp. 117-119) shows that, when they exist, limits of
the (1 - a) confidence region are given by
-
y
x' =X + ---;;:- +
af31
-)
y2- a[t2MSe0 + liJ)-
y2]
/\
af31
(10-4a)
and
y -j
-
X"= X +---;;:--
af31
where
gi'L ex
i-
y2]
(10-4b)
af31
t2Ms
a= - -1
--_.:;_t: _ _
y2- a[t2MSe0 + liJ)"
-
1 and t = t( a/2, I - 2), the t- tatistic.
f)2
i=l
xz0 , xhi• or both may not exist. This is because, if the line slope could be zero at the
required level of confidence, the X intercept may be at infinity. That is, if the OLS
regression line could be parallel to the X axis, the intercept will be unbounded.
Consequently,
only meaningful confidence intervals can be obtained from Eqs. (10-4) if
/1.
/3 1 is significantly nonzero.
Montgomery and Peck (1982, p. 402) and Seber (1977, p. 189) discuss another
approach for estimating the X intercept and its confidence interval. If the explanatory
variable is also a random variable (i.e., not a controllable variable), an X-on-Y regression
could be performed to estimate the X intercept and its confidence interval. They suggest
that the method ofEqs. (10-4) is generally preferred but there is not much difference when
R2 is large.
Example 5- Estimating Grain-Density Variability from Core and Wireline
Data. The bulk-density log is often used to estimate formation porosity if
certain parameters of the lithology are known. Conversion of the bulk
density (ph) to porosity (¢D) requires the densities of the pore fluid (pfl) and
grains (PmcJ be known or estimated:
For this model, Pb = Pma when ¢n =0. If Pb is plotted against porosity
measured by another method (l/J plug), the linear model may apply and Pma
estimated from the Ph-axis intercept.
Statistics for Petroleum Engineers and Geoscientists
251
Figure 10-8 shows a scatter plot of core-plug porosities and the corresponding
wireline density measurements for a well. A strong linear relation appears,
despite the large difference in volumes of investigation. If the porosity
heterogeneity were large at the density-log scale, this relation might have
been much weaker. The data here suggest that Pb and the model could form a
useful porosity predictor. As a check on the quality of the line fit, however,
the Pb intercept <Pma) should match the grain density measured from core
samples <Pmad· In this case, an average of laboratory measurements of core
samples gives PmaC = 2.67 g/cm3 , while Pma = 2.69 g/cm 3. Is this
difference significant at the 95% level?
0.32
<l>plug = -0.569pb + 1.530
R 2 = 0.915
0.28
<!>plug
0.24
0.20
2.15
2.20
2.25
2.30
2.35
pb, g/cm3
Figure 10-8.
Wireline density versus core-plug porosity scatter plot and
regression line.
For this data set, we have I= 20, t(0.025, 18) = 2.101, X= 2.246,
"'L(Xi- X) 2 = 0.0256, Y = 0.252, MSt: = 0.0000425, and 1 = -0.569.
Using Eqs. (10-4), these data give a= -0.977, xz0 = 2.63, and xhi = 2.77.
Thus, the results PmaC = 2.67 and Pma = 2.69 are not significantly different
at the 95% level, since PmaC falls within the 95% confidence interval for Pma·
/3
Since neither ¢plug nor Pb are controlled variables, we can regress Pb on ¢plug
to obtain another estimate of Pma· These data give
= -1.61¢plug + 2.65,
implying that Pma = 2.65 g/cc. The 95% confidence interval for this Pma is
Pb
252
Chap. 10 Bivariate Analysis: Further.Linear Regression
2.65 ± 2.101(0.029) or 2.59 to 2.71, calculated using Eq. (9-7a). This interval
is slightly smaller than the one obtained using Eqs. (10-4), but both methods
suggest the core and log matrix densities are similar.
In the above example, the 95% confidence region does not lie symetrically about the X
intercept: 12.63 - 2.691 =1- 12.69 - 2.771. This is reasonable since ~ 1 - fh = 8fh is equally
likely to be positive or negative, but the commensurate change in the X -axis inte~em;,
OX, is nonlinearly related to 8{3 1. Because the line must pass through the point ( X, Y),
by geometry we have that 8X = - Y8{3 1![{3 1({3 1 + 8{3 1)]. Thus, except for very small
8{3 1 , 8X and 8{31 are nonlinearly related and, consequently, an asymmetrical confidence
interval occurs.
Another situation involving discrimination is the definition of nonproductive intervals.
Frequently, nonproductive portions of the formation are defined on the basis of
permeability. For example, 1 mD is a common cut-off value with only intervals having
permeabilities above that value being declared as "net pay" for liquid flow. As described
in Example 1, however, permeability is difficult to measure in situ, so porosity is often
used to predict the permeability. In the following brief example, we obtain the best
estimate of porosity for the cutoff value and its range of uncertainty.
Example 6- Defining the Net Pay Cutoff Value Using a PorosityPermeability Correlation. We use the data set presented in Example 2 and
interpret Y as the permeability and X as porosity. We assume that a cutoff
value of approximately 1 mD is required. We use the term "approximately"
here because the bias of the line may also be a consideration. In this case,
log(l) = 0 actually represents 1.5 mD permeability. However, as we see
below, the bias of the predictor is an unimportant issue here.
In this case, the X axis (log(Y) = 0) is the desired cutoff value, so the
/1.
/1.
porosity corresponding to this permeability is -/30//31 = 1.41/15.4 = 0.092.
Eqs. (10-4) provide the 95% confidence interval endpoints as X"= 0.144 and
X'= 0.037. In other words, an interval with permeability 1 mD has a 95%
probability that its porosity will fall somewhere in the interval [0.037,
0.144].
It is surprising that even with nearly 100 data points and a strong porosity-
permeability correlation, this example gives a wide porosity interval. The
variation in possible X intercept values represents approximately two orders
of magnitude in permeability variation. This large variability is why the
estimator bias is of relatively little importance.
Statistics for Petroleum Engineers and Geoscientists
253
The primary source for this X intercept variability can be explained by
thinking of the porosity variability about the line for a fixed permeability.
The standard deviation of the porosity is about 0.05 and the line, having
R 2 = 0.79, reduces this variation to about O.os--J 1-0.79 = 0.023 at a fixed
permeability. Multiplying this variation by t = 2 for the 95% interval
suggests the precise porosity for a given permeability is not going to be
pinned down to better than ±0.05 or so unlesss we can include more
information into the prediction.
Examples 5 and 6 differ in an important aspect. The confidence interval of Example 5
is genuinely used in only one test to compare the X intercept with a value determined by
another method. The confidence interval determined in Example 6, however, might not
be used just once; it could be applied on a foot-by-foot basis across the entire interval of
interest. In this case, if a type 1 error at the 95% level is to be avoided for all intervals
simultaneously, a modified form of Eqs. (10-4) needs to be used. Miller (1981, pp.
117-119) has the details.
1 0~6 IMPROVING CORRElATIONS FOR BETTER
PREDICTION
In several examples of this and other chapters, we allude to a common problem in
reservoir characterization. Frequently, we attempt to relate measurements having very
different volumes of investigation. In homogeneous materials, this would not be a
problem. Inheterogeneous media such as sedimentary rocks, however, differences in
volumes of investigation tend to weaken relationships among measurements.
Reconciling and comparing different measurements on different volumes of rock is a
broad and difficult area. Much research remains to be done. We can, however, give a few
tips and illustrate the ideas with an example.
Relationships between measurements are stronger when
1.
2.
3.
there is a physically based reason for the two quantities to be related (e.g.,
porosity and bulk density, resistivity and permeability, radioactivity and clay
content);
the measurement volumes are similar (e.g., core-plug porosity and permeability,
probe permeability and microscale resistivity, grain size and probe permeability);
and
the measurements have sample volumes consistent with one of the geological
scales (e.g., probe permeability and laminations, resistivity and beds, transient
tests and bed sets).
254
Chap. i 0 Bivariate Analysis: Further Linear Regression
The first item avoids exploiting "happenstance" relationships that may disappear without
warning. The second item ensures that, in heterogeneous rocks, one measurement is not
responding to material to which the other sensor cannot respond, or responds differently.
The third item exploits the semirepetitive nature of geologic events by making
measurements on similar rock units.
Example 7 -Assessing Probe Permeameter Calibrationfrom Core-Plug Data.
While plug and probe permeameters both measure permeability, their
volumes of investigation and the boundary conditions of the measurements
differ considerably. This would suggest that there is not necessarily a one-toone correspondence between the measurements.
8
40
6
30
"""'~
1-<
ClJ
,aQ. 4
J:l
s
z=
~
'-"'
..5
2
20
10
ln(k
0
0
2
0
4
ln(k probe)
Figure 10-9.
6
8
0.0
0.4
0.8
Cv
Scatter plot and regression line (left) and histogram of
probe-based variability (right) of Rannoch formation core
plugs.
Figure 10-9 left is a plot of probe and plug data taken from the Rannoch
formation (discussed in Example 6 of Chap. 3). Each core plug had four
probe measurements on each face, giving eight probe permeabilities per plug.
Some plugs showed considerable variability; Fig. 10-9 right is a histogram of
the probe-based Cy's from each plug, in which we see that 29% of the plugs
have Cv;?; 0.5. this variability arises because each core plug is sufficiently
large to contain several laminations, which the probe is detecting
individually.
A plot of probe-measured variability (Fig. 10-10 left) supports this observation. The average and ±1 standard deviation points are shown for the Cy's of
1.2
Statistics for Petroleum Engineers and Geoscientists
255
three rock types common to the Rannach formation. Rocks with stronger
visual heterogeneity tend to exhibit greater permeability variation.
A regression line using all the data (Fig. 10-9 left) suggests that the probe
and plug data are systematically different(i.e., 0.84 =1= 1 and 0.79 =1= 0). Some
of this difference, however, may arise because of the highly heterogeneous
nature of some samples. A regression line using only the plugs with
Cv::;; 0.4 shows the probe and plug measurements are similar and, therefore,
no further calibration is required; the slope and intercept are nearly 1 and 0,
respectively. Thus, there is good cause for some probe and plug data to differ,
and the cause here lies with the scales of geological variation to which each
measurement is responding.
1
0.8
0.6
0.4
0.2
Cv
8
l
1
+
Plugs with Cv ::;; 0.4
6
,-._
1+
l
~
T
+
l
0
"'
iS..
0
.5
4
2
R 2 = 0.95; N = 45 plugs
) = 0.98ln(k probe ) + 0.09
0
Lam.
WklyLam.
Mass.
0
Rock Type
2
4
6
8
ln(kprobe)
Figure 10-10. Probe permeability variation observed as a function of
visually assessed rock type (left) and a probe-plug
permeability scatter plot (right) for plugs with limited
variability. Lam.= laminated, Wkly lam.= weakly
laminated, and Mass. = massive.
1 0=7 MULTIVARIATE REGRESSION
As explained in the introduction to Chap. 9, multivariate linear regression is not covered
in detail in this book. Nonetheless, we will say a few words about it because it is very
common and we will use many of its concepts when we discuss Kriging in Chap. 12.
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Chap. i 0 Bivariate Analysis: Further linear Regression
The model now is
M
Y
=f3o + L f3mx(m) + e
(10-5)
m=l
Y
where x<m) is a set of the observed data {Xlm), x<-t), ... , x<'F)}, and
is a set
containing estimates of the variable to be predicted. e is the white noise whose elements
have zero mean and are uncorrelated with anything except themselves. The procedure is
the same as before; we take the expectation ofEq. (10-5) to solve for f3o·
M
f3o = E(Y) -
L f3mE(X(m))
(10-6)
m=l
After inserting the model and eliminating f3o with Eq. (10-6), we have
M
Var(Y-
M M
h = Var(Y)- 2 L f3mCov(Y, x(m)) + L L f3nf3mCov(X(n), x<m))
m=l
n=l m=l
This expression is the same as the sum-of-the-squares operatorS given in Chap. 9 if we
replace the expectations with their estimates. To find the regression coefficients, set the
derivative with respect to each regression coefficient to zero:
as as
as
a{31 = afh = ... = a{3M = O
These operations lead to the following normal form of the multivariate linear model:
=
(10-7)
where we have substituted the covariance definitions for the expectation operators.
Cx<l)x(M) = Cov(xCl), x(M)), etc. Equation (10-7) represents a set of M linear
Statistics for Petroleum Engineers and Geoscientists
257
equations that can be solved by standard means. The coefficient [30 follows from Eq.
(10-6) after determining {31 through f3M·
The matrix on the left ofEq. (10-7) is the covariance matrix; its elements represent the
degree of association between the "independent" variables x(l) through x(M). If the
variables are independent (of each other), the matrix is diagonal and the resulting solution
much easier. In any event, the diagonal elements are the variances of each variable.
Example 8a - Multivariate Regression to Examine Relationship of Oil
Recovery to Slug Size and Pressure. We can illustrate the above procedure
and introduce two other statistical insights with a fairly simple example.
The following table represents the incremental oil recovery (Y) from seven
immiscible-gas-injection projects (adapted from Quigley, 1984) and their
corresponding slug size (XCl)) and average pressure (XC2)).
Incremental
Oil Recovery,
fraction
0.394
0.06
0.4
0.189
0.091
0.0997
0.128
Slug Size,
fraction
0.3742
0.00157
1.515
0.04707
.........
0.00002
0.0031
0.5764
_._....._......_............._
Average Pressure,
psig
!
3671
1695
4046
4200
2285
2049
1050
For this problem, M = 2 and I= 7. We seek the model coefficients {3 0 , {3 1,
and {32 .
For numerical reasons, it is often advantageous to work in standardized
normal variates, particularly when the range of data is large as it is here. To
do this we require for Y
y = (0.394 + 0.06 + ... + 0.128)!7 = 0.195
and
(jf = [(0.394)2 + (0.06)2 + ... + (0.128)2]/7 - (0.195)2 = 0.0207
from which we have, for example,
Chap. 10 Bivariate Analysis: Further Linear Regression
258
Y1 = (0.394- 0.195)/(0.0207)112 = 1.386 and similarly for the other values
and for x*(l) and x*(2). The following table shows the standardized variates.
y*
x*O)
0.02615
-0.6423
2.073
-0.5607
-0.6451
-0.6396
...................
0.3889
1.386
-0.9347
1.428
-0.0384
-0.7193
-0.6589
-0.4622
·-----------~~~
~~
x*C2)
0.7676
-0.8168
1.068
1.191
-0.3438
-0.533
-1.334
--
..........>..>..Vo..U~-~-
~~A.O.-...
---
Such a transformation also renders the model equation dimensionless. {30 = 0
in the standardized variates, since the expectations of the new variables are 0.
Also, their variances are 1.
The normal equations require the covariances; an example calculation for one
element is
Cov(X*(l), x*(2))= [(0.02615)(0.7676) + (-0.6423)(-0.8168) + · · · ]/7
= 0.3049
which leads to the following normal equations:
[
1
0.3049
0.3049
1
J[ {31f32 J =[ 0.6176
J
0.6460
Owing to the standardization of the variables, the elements of the matrix on
the right side are now correlation coefficients. The solution to these leads to
the following regression equation:
y*
= 0.4637 x*Cl) + 0.5046 x*C2)
A
Apparently, Y is about equally sensitive to both X*(l) and x*(2). However,
x*(l) and x*(2) are somewhat correlated.
Physical arguments on this type of process suggest that incremental oil
recovery (IOR) should increase roughly linearly with slug size as the model
259
Statistics for Petroleum Engineers and Geoscientists
suggests (Lake, 1989, Chap. 7). The dependency of lOR on pressure is more
complex, but the two should be positively correlated also. Moreover, there is
no obvious reason for slug size and pressure to be correlated; hence, the
0.3049 correlation coefficient is likely to be not significant. Of course, from
a statistical point of view, we should interrogate e for independence and
normality (Chap. 9).
Analogous to the analysis for bivariate models, we can obtain confidence intervals for
the parameter estimates of multivariate regression. This topic is covered in Montgomery
and Peck (1982) and other regression texts.
If the covariance (correlation) matrix is nonsingular, it is possible to redefine the
variables again to make them independent. We need a little development to see this.
Write the original model as
1\
f3
Y = xT
and the normal equations as
Cxx
f3 = Cxy
in matrix form. For any nonsingular matrix P we can write
f3
Cxx p-lp
= Cxy
It is also possible to multiply both sides by P to arrive at
- - -
Cxx f3= Cxy
-
whereCxx =P Cxx p- 1 ,
problem
-f3 = P/3, and Cxy
- =PCxy
1\
-
y= X
in the new variable X
all define a regression
f3
= X P -1.
-
The new covariance matrix Cxx will be diagonal if the columns of the matrix P
consist of eigenvectors of the original covariance matrix Cxx·
Chap. 10 Bivariate Analysis: Further Linear Regression
260
Example 8b -Multivariate Regression to Examine Relationship of Oil
Recovery to Slug Size and Pressure. For the case here, P and p-l are
both
p = [ 0.7071 0.7071
0.7071 -0.7071
The new variables are
x(l')
and
x<2')
J
= 0.7071 x(l) + 0.1011 xC2)
= 0.1011 x(l) - 0.7071 xC2)
and the new regression equation is
-Y = o.6847 x<n - o.o289 xC2')
where the variables on the right side are now independent.
The principal reasons for going through this is that the variance of Y can
now be attributed individually to the new variables. Using Eq. (4-3), we
have
Var {Y) = (0.6847)2 Var(XO')) + (-0.0289)2 Var(X(2'))
or 63.56 % of the variance of
Ycan be attributed to x<l'), only 0.05% to
x<2'), and the remainder to e. Variance analysis on redefined variables
such as this is called factor analysis, a topic covered in more detail in Carr
(1995, pp. 99-102).
1 0=8 CONClUDING
COMMENTS
Linear least-squares regression is a relatively simple method for developing predictors.
Despite this mathematical simplicity, the method has a wealth of statistical features and
interpretations that can make it enormously powerful. The material-balance analysis
(Example 4 of this chapter) is a good example of this. Simple OLS regression produced
estimates of m and Nand no statistical analysis was needed to obtain these results. The
data, however, have more information that can be interrogated to produce estimated
confidence levels for m and N. These precisions have quite an influence on the way we
look at the resulting m and N values. Without them, N = 105 million stock tank barrels
Statistics for Petroleum Engineers and Geoscientists
261
(MSTB). With them, we see that N is about 105 MSTB, give or take 34 MSTB, and
that we need more data to reduce the uncertainty to a tolerable level, dictated by the field
economics.
Chapter 9 and this chapter represent some of our experiences with regression. Much
more could be said and there are excellent books on the subject, which we have cited. We
encourage the reader to explore these aspects further and expand their "regression tool kit."
Regression· is a fascinating area that seems limited only by the user's ingenuity.
11
ANAL YS/5 OF SPATIAL
RELATIONSHIPS
We have now spent three chapters exposing aspects of correlation among different random
variables. In addition, in Chap. 6 we discussed heterogeneity and its measures. A more
complete exposition of ways to represent spatial structure requires information about both
heterogeneity and correlation, except now this is the correlation of a random variable with
itself. Such autocorrelation (or self correlation) is the subject of this chapter.
Correlation is the study of how two or more random variables are related. In data
analysis, there are three types of correlation, as depicted in Fig. 11-1.
1.
Simple correlation, such as between permeability and porosity in Well No. 1 in
Fig. 11-1, is the relationship between two variables taken from two different
sample spaces but representing the same location in the reservoir. The analysis
and quantification of this type of relationship were considered in Chap. 8.
2.
Crosscorrelation, such as of permeability between Well No. 1 and Well No. 2 in
Fig. 11-1, is the relationship between two samples taken from the same sample
space but at different locations. Examining electrical logs from several wells for
263
Chap. 11
264
Analysis Of Spatial Relationships
characteristic patterns-the generic usage of "correlation" in petroleum engineeringis one example of crosscorrelation.
3.
The last type of correlation, autocorrelation, is between variables of the same
sample space taken at different locations. This is illustrated on the permeability
data of Well No. 1 in Fig. 11-1. This correlation addresses, for example, the
sequence of lithofacies that appear in a well. The last two types of correlation are
both autocorrelation but in different directions.
Well No.1
1,o-21o-1 1po 1p1 1P21o3
Permeability, mD
Figure 11-1.
Well No.2
Well No.1
0102030
10"2 10" 1 10° 10 1 10 2 10 3
i
I
I
I
I
Permeability, mD
Schematic illustration of correlation types for porosity and
permeability.
I
265
Statistics for Petroleum Engineers and Geoscientists
Because time-dependent data are usually in the form of series of data, sometimes
autocorrelation and crosscorrelation fall under the subject of time-series analysis. In what
follows, we will not make a distinction between data from time-sampled sequences and
data from spatially sampled seguences.
11·1 DATA TYPES
Recall from Chap. 2 that there are two types of measurements recorded in series, strings,
or chains.
1.
Continuous variables recorded on a known or measurable scale, e.g.,
(a) electric logs measuring resistivity as a continuous function of depth,
(b) production history or time-dependence of an individual well, and
(c) core-plug data recorded as a function of specified depth locations.
2.
Categorical or discrete variables with/without a known or measurable scale, e.g.,
(a) lithologic states recorded in a sedimentary succession,
(b) the occurrence of certain types of microfossils as functions of depth, and
(c) successions of mineral types encountered at equally spaced points on a
transect across a thin-section.
These measurements may be regular (equally spaced) or irregular. The type of variable
and the spacing affect the method and extent of analysis possible, as Fig. 11-2 indicates.
Regularly Spaced Samples Irregularly Spaced Samples
Interval or
Ratio Data
1.
2.
3.
4.
Nominal or
Ordinal Data
1.
2.
3.
4.
Figure 11-2.
Regression
Time-trend analysis
Autocorrelation or
semivariograms
Spectral an.alysis
Auto association or crosscorrelation
Markov chains
Runs test
Indicator semivariograms
2.
3.
Regularization
Regression
Autocorrelation or
semivariograms
1.
Series of events
1.
Summary of selected relationship modeling and assessment methods
according to the nature and spacing of the sampled variable. Adapted from
Davis (1973).
In the following, we discuss a few of these techniques, concentrating primarily on
·
regularly spaced data.
266
Chap. 11
Analysis Of Spatial Relationships
11-2 NOMINAL VARIABLES
A run is a successive occurrence of a given discrete variable, which includes a single
occurrence. This is the simplest type of sequence because an ordered set of qualitative
observations falls under two mutually exclusive states. A· test for randomness
(independence) of occ\rrrence is formulated from the number of runs of a given state. We
can determine the probability of a given sequence or run being created by the random
occurrence of two states. We do this by enumerating all possible ways of arranging n 1
items of state 1 and n2 items of state 2. Taking u to be the total number of runs in a
sequence and assuming n 1 and n2 are greater than 10, u is approximately normally
distributed (Fuller, 1950, pp. 194-195). The expected value is
2n 1n2
f.L ~ E(u) = 1 + --=-=-
nl
u
+ n2
with variance
Knowing the mean and variance of u, we can apply
version:
a hypothesis test to its normalized
u - f.Lu
z=-Cfu
Example 1 -Illustration of Runs Test. Analysis of an outcrop section
revealed the following. sequence of rock types sampled every 30 em along a
line:
QQSQQQSQQQSSQQSQQQSSQSQQQSQSS
where S =shale and Q =sandstone. The data are discrete variables that, being
geologic features, are usually grouped or autocorrelated. The grouping or
autocorrelation could take one of two forms: either there may be fewer
transitions (positive autocorrelation) m: there may be more transitions
(negative autocorrelation or anticorrelation) than the expected number for a
random selection of the S's and Q's. We can detect either case using a runs
test.
The null and alternative hypotheses in this case are
H o: u
=f.Lu (i.e., no autocorrelation)
Statistics for Petroleum Engineers and Geoscientists
267
HA: u #= J.lu (i.e., positive or negative autocorrelation).
Hence, at a 5% level of significance, lzl > 1.96 would lead to rejection of H 0 .
The statistics suggested by the above are
n 1 = number of shale points = 11
n2 = number of sandstone points = 18
J.lu =expected number of runs= 14.7
a~
u =observed number of runs= 16
=variance of u =6.2
u - J.1
z = ___u = 0.5 < 1.96
O"u
So, we accept Ho in favor of HA at the 5% level of confidence. We could
analyze this set further to determine the existence of autocorrelation, but the
runs test indicates no structure ..
The runs test is a simple test for the presence of autocorrelation. In many problems,
however, such a test does not provide enough information to permit successful
exploitation of the answer it gives. This failing is primarily because of the qualitative
nature of discrete variables. For interval or ratio variables, more powerful techniques are
possible. We discuss these next.
11-3 MEASURES OF AUTOCORRElATION
Recall from Chap. 8 that the covariance, Cov(X, Y), is a measure of the strength of a
relationship between two variables X and Y. In that case, nothing was mentioned about
the spatial locations of X and Y because we assumed that both reflected properties at the
same location. In a similar way, we can use the covariance to measure the relationship
strength between a variable and itself, but at two different locations.
Let Z denote some property (e.g., permeability or porosity) of the rock that has been
measured on a regular grid (such as along a core) as sketched in Fig. 11-3. Zi is the
property Z at location Xi, or Zi = Z(xi). The location xis a time-like variable (nonnegative and always increasing with i). L!Jh =Xi+ 1 -Xi is the spacing or class size.
Number
1
2
3
Location
xl
X2
X3
Value
zl
z2
z3
Figure 11-3.
i+l
I
Xi
Xi+l
XJ
Zi
Zi+l
ZJ
Schematic and nomenclature of a property Z measured at
points along a line.
268
Chap. 11
Analysis Of Spatial Relationships
Autocovariance/Autocorrelation
We can now define the autocovariance between data Zi and Zj as
Cov(Zi, Zj) = E ( [Zi - E(Zi)] [Zj- E(Zj)]} = E(ZiZj) - E(Zi) E(Zj)
(11-1)
The quantity Cov(Zi, Zj) is a measure of how strongly the property Z at location Xi is
related to Z at location Xj. When Xi and Xj are close together, we expect a strong
relationship. As Xi and Xj move farther apart, the relationship should weaken. The
properties of the autocovariance are similar to the covariance discussed in Chap. 8,
specifically Cov(Zj, Zj) = Cov(Zj. Zj) and Cov(Zi, Zi) = Var(Zi).
The autocovariance, being a generalized variance, is a measure of both autocorrelation
and variability (i.e., heterogeneity). To remove the heterogeneity, we define an
autocorrelation coefficient between Zi and Zj as
As in the case of the correlation coefficient discussed in Chap. 8, p must be between -1
and +1. Many times both Cov and p are written in terms of a lag or separation index k,
where j = i+k.
The similarities between the definitions of autocovariance and covariance belies a
subtlety that has significance throughout the remainder of this discussion. Recall that the
basic definition for the expectation operator involved a probability density function (PDF)
for the random variable (since we are dealing with two variables here, these are actually
joint PDF's). Taken to Eq. (11-1), this means that Zi and Zj both must have PDF's
associated with them even though they are from the same parameter space. Taken as a
single line of data, Fig. 11-3, a PDF at each point is difficult to imagine. If we view the
line as only one of several possible lines, however, things begin to make more sense
because we can imagine several values of Z existing at a point Xi. If the PDF's for all Zi
are the same, we have assumed strict stationarity.
A related hypothesis comes from the need to estimate Cov. Equation (11-1) contains
E(Zj), which we would normally estimate with a formula like
1 I
E(Z) =-
L
I i=l
Zi
Statistics for Petroleum Engineers and Geoscientists
269
It is not clear that E(Z) estimated in this fashion (the spatial average) would be the
same as if we had averaged all the values at Xi from the multiple lines (an ensemble
average). Assuming an equality between spatial and ensemble averages is the ergodic
hypothesis. We assume ergodicity and some form of stationarity in all that we do.
Example 2- Variance of the Sample Mean (Revisited). We recalculate the
variance of the sample mean, as was done in Example 2 of Chap. 4, but now
allow the random variable to be dependent. As before, we have
Inserting the definition for variance and multiplying out yields
. 1
Var(Z) = p {E[Z1(Z1+···+ ZI) +···+ZI(Zl +···+ ZI)]-
[E(Z1)(E(Z1) +···+ E(ZI) +···+ E(ZI)(E(Zl) +···+ E(ZI)]}
We can no longer drop the cross terms. Further multiplying and
identification with the Cov definition yields
_
Var(Z)
l
=2
I
I
I
I, I,Cov(Zi.Zj)
i=l j=l
an equation with a considerably different form from the case with the Zi
independent. The inclusion of autocorrelation may increase or decrease the
variance of the sample mean, compared to the independent variable case, since
Cov can be either positive or negative.
The variance of the sample mean has many different forms. For example,
_
Var (Z)
2I
I
)
=~2
I 1 + 1 .l . ~p(Zi.Zj)
l=l J=z+l
where Var(Zi) = (j2. This equation is the starting point for the study of
dispersion in an autocorrelated medium or of any similar additive property.
Chap. 11
270
Analysis Of Spatial Relationships
Semivariance
A common tool in spatial statistics is the semivariance y, defined as
(11-2)
The semivariance is also a generalized variance; compare its definition to Var(Z) =
E[(Z- z)2]. It shares many properties with the variance, most notably that it is always
greater than zero. However both Cov and y are limited in utility without further
assumptions, which we go into below. Sometimes, a normalized version of the
semivariogram is considered:
Stationarity
One of the consequences of strict stationarity (discussed above) is that all moments are
invariant under translation, that is, they are independent of position. Stationary of order r
means that moments up to order r are independent of position. Since we do not use
moments higher than 2, all we require for the moments is second-order stationarity, but
even this has some subtle aspects.
We have already used second-order stationarity in Example 2, where we set
Var(Zi) = a2, or
which is free of an index on the right side. Also
from which it follows that Cov(Z 1, Z1 +k) = Cov(Z2, Z2+k), etc. (Second-order
stationarity implies that first moments are also independent of position.) Thus, secondorder stationarity renders the autorrelation measures independent of position:
Cov(Zi, Zi+k)
= Cov(k) = Covk
p(Zi. Zi+k) = p(k) = Pk
Statistics for Petroleum Engineers and Geoscientists
271
or, the autocorrelation measures become functions of a single argument. In terms of the
line-of-data viewpoint (Fig. 11-3), these become
Cov(k) = Cov(k-1.h)
= Cov(h)
p(k) = p(Mh) = p(h)
y(k) = y(k-1.h) = y(h)
where h (k-1.h) is the separation distance, lag distance, or sometimes just lag.
It is difficult to overstate the simplifications that second-order stationarity brings to the
autocorrelation measures. For one thing, there is now a simple relationship between the
autocovariance and the semivariance,
y(h) = a2 - Cov(h)
(11-3)
which holds if the variance is finite. A second advantage is that there is now possible a
graphical representation of the autocorrelation measures. A plot of Cov versus h is called
the autocovariogram, yversus h the semivariogram (almost always referred to as the
variogram), and p versus h the autocorrelogram. The qualitative behavior of these is
discussed in the following paragraphs.
Figure 11-4 illustrates an autocorrelation coefficient p decreasing with lag distance
(k or h).
When p approaches zero there is, on the average, no correlation between data at that
separation. The lag where this occurs is a measure of the extent of autocorrelation. The
left curve shows a modest amount of autocorrelation, the center curve shows even less,
and the right curve shows one with a very long autocorrelation distance. This last figure
indicates a trend or a spatially varying mean that would probably be apparent from
carefully inspecting the original data set.
Another form of stationarity is sometimes invoked when using the semivariogram. It
is slightly weaker than second-order stationarity. If the difference Zi- Zi+k has a mean
and variance that exist and are stationary, then Z satisfies the intrinsic hypothesis. It is
weaker because Var(Z) may not exist.
Chap. 11
272
Analysis Of Spatial Relationships
It is often assumed that second-order stationarity applies. If so, Eq. (11-3) implies that
p and yare mirror images of each other. When p decreases, yincreases. If p(h) < 0 for
some h, y(h) > a2. Thus, anticorrelation appears as an increase in yabove the variance.
1
1
1
Pk
0
-1,_--,---.---.---·-l~--.----.--.---,_1~--~--~--~--~
0
5
10
15
20
0
5
10
15
20
0
5
10
15
20
Lag Index, k
Figure 11-4.
Schematic of autocorrelogram with modest (left), little
(center), and strong (right) correlation.
As beneficial as they are, ergodicity and stationarity are difficult to verify in a set of
data. Verifying ergodicity would mean measuring multiple lines and seeing if ensemble
statistics compare with spatial statistics. However, any such differences would be
indistinguishable from heterogeneity. We could, similarly, calculate Cov and ywith
successively smaller sets of the data but, at some point, we would be unable to conclude
anything because the number of data would be small.
Another subtlety of yand pis that, because they are related to the autocovariance, they
must obey certain properties. The most important is that p must be a positive
semidefinite function and, correspondingly, ymust be a negative semidefinite function.
Since the mathematical terms positive and negative semidefinite may not be familiar to
the reader, we will discuss these properties further by examining the implications for the
autocorrelation p.
The term positive definite is for functions what the requirement to be positive is for
scalars (A> 0). Semidefiniteness corresponds to the scalar requirement of nonnegativity,
A~ 0. Because we are dealing with functions, however, positive semidefiniteness is
more complicated. For the autocorrelation function, p, it means that the correlation at
one point, Xi, is related to the correlation at any other point, Xi+k· For example, we will
Statistics for Petroleum Engineers and Geoscientists
273
see later that any interpolation procedure requires that the estimated value at the new
location be a linear combination of the values at the I measured locations:
I
Z* = I, AiZi
i=l
where the weights (A.'s) are determined by a specific procedure. If I= 2, we have
Z* = A- 1z1 + A:2Z2. Since Z* is a random variable, it has a variance given by
If Z is stationary,
Var(Z*) = Var(Z)
[AJ:2 + 1t22 + 2A.l ~p(l)]
[A,; A.;
Since Var(Z*);::.:: 0, we require that
+
+ 2A. 1 A.2p(l)] ;: .: 0. This can only be
achieved for any A- 1 and A-2 if p(l) ::::; 1. Clearly, when I> 2 the situation is analogous
but more complicated. This is where the positive semidefiniteness of p is needed: the
Var(Z*) must be nonnegative for any values the A.'s take. Thus, not just any function
can be used for p; it has to be positive semidefinite. This limits the possibilities quite
considerably, and several examples will be considered below.
11-4 ESTIMATING AUTOCORRELATION MEASURES
Like all statistics, Cov, p, and ymust be estimated. We discuss here only the more
common estimators. See Priestley (1981, Sec. 5.3) and Cressie (1991, Sec. 2.4) for
further estimators and their properties.
Autocovariance/ Autocorrelation Coefficient
The most common estimator for the autocovariance between data points Zi and Zi+k is
Chap. 11
274
-
Analysis Of Spatial Relationships
A
where Z is an estimate for the mean value of Z and C is an estimate for Cov.
Subject to the assumption of second-order stationarity (that Cov depends only on the
separation index k), this becomes
(11-4)
If the separation distance or lag is zero, the above becomes an estimator of the variance
that itself becomes an appropriate normalization factor. An estimator for the
autocorrelation coefficient is
(ll-5)
the denominator being, of course, an estimate of the variance. The estimator in
Eq. (11-5), though common, may not be a positive semidefinite function; see Priestly
(1981, p. 331).
Example 3 -Autocorrelation of an Ordered Card Deck. Let us calculate the
autocorrelogram for a standard deck of cards (four suites of ace through king
each). This is also a concrete way to illustrate autocorrelation, for we can
show that "randomness" in card decks really means absence of autocorrelation.
Let ace be one, etc., up to king equaling 13. The mean value in the deck is 7
and the standard deviation 3 .78. Neither of these statistics expresses the
actual spatial ordering, however.
A completely ordered deck has an autocorrelogram that looks something like
Fig. 11-5 left. There is perfect autocorrelation at every thirteenth card, as
expected, and the pattern repeats itself. This means that, given the value and
position of any card in the deck, the value of any card that is 13n (n an
integer) away has the same value. Every seventh card shows very strong
negative autocorrelation since these separations are, on average, as far apart as
possible in numerical value. In other words, given the position p of a card
with value Vp, if vp> 7 (the mean value of the deck), the card at position
p + 7 is likely to have Vp+? < 7 (and vice versa).
Statistics for Petroleum Engineers and Geoscientists
0.0
1.0
0
0
0
0.5
Pk
0
0
1.0
275
0.5
0
..... .9.................~..........~.................9............. .
0.0
-0.5
-0.5
-1.0
-1.0
0
20
10
30
0
Figure 11-5.
10
20
30
Lag Index, k
Lag Index, k
The autocorrelogram of an ordered card deck. Originally
ordered (left); after one to three shuffles (right).
Figure 11-5 right shows the autocorrelograms after several shuffles of the card
deck. The autocorrelation is much more erratic (the shuffle was not perfect)
but, after the first shuffle, there is a hint of periodicity at every 26th card.
More importantly, the autocorrelation coefficient approaches zero only
gradually.
After three shuffles, there is very little dependence on adjacent cards. Besides
the importance of not shuffling card decks fewer than three times when
gambling, this example shows that autocorrelation is a useful tool for
evaluating spatial dependencies.
Semivari ograms
The standard estimator for the semivariance is
2Yk = 1
~
1-k
k
L (Zi - Zi+k) 2
i=l
(11-6)
Chap. 11
276
Analysis Of Spatial Relationships
This formula expresses the variance of I - k sample increments measured a distance k
units apart, again assuming second-order stationarity. The sample semivariogram may
look something like Fig. 11-6.
The data points in the sample semivariogram rise with lag, although not necessarily
smoothly or monotonically, reflecting greater heterogeneity for larger sample scales. The
data may or may not tend to a plateau as the lag increases; similarly, they may or may
not tend to zero as the lag decreases. We quantify the shape of these trends in the model
curves discussed later in the chapter.
(!)
u
0.4-
c::
m
·a::
.~ 0.2
E
~
0
0
1
2
3
4
5
6
7
8
lag,k.1h
Figure 11-6.
Schematic illustration of a sample semivariogram.
Although it is not their prime purpose, both the semivariogram and the
autocorrelogram can be used to detect periodicities. As Example 3 showed, large values
of p fork> 0 may reflect a repetitive feature in the data. Similarly, repetitiveness with a
semivariogram is shown by a reduction in yfor k > 0, This is called a hole. Some
examples will be considered next.
11-5 SEMIVARIOGRAMS IN HYPOTHETICAl MEDIA
Simple synthetic permeability trends are useful to demonstrate the form of the
semivariogram in typical geological scenarios and to demonstrate what can or r'1nnot be
extracted in their interpretation.
Statistics for Petroleum Engineers and Geoscientists
277
Large-Scale Trends
Detenninistic trends (i.e., with no random component) produce a semivariogram function
that is monotonically increasing with increasing lag. Both linear trends (Fig. 11-7A) and
step changes (Fig. 11-7B) are common depositional features, although they are rarely as
easy to recognize in practice as is suggested by these figures. Cyclic patterns are also
common.
Distance
4000
10
8
Experimental
Semivariance,
mo2
3000
6
2000
4
IS
1000
2
II
Ill
0
0
250
500
Permeability, mD
Figure 11-7A.
0
1
2
3
4
5
6
Lag
Sample semivariogram for data with a linear trend.
Both features give similar semivariograms; the form of the semivariogram only
indicates an underlying large-scale trend. It provides no insight into the fonn of that
trend. Indeed if the simple trends were inverted (reversed), the resulting semivariograms
would be indistinguishable.
Chap. 11
278
Experimental
Semivariance,
Distance
10
.12
.08
6
'I
.06
I
.04
411>
•
2
/
Variance
Ill
''
I
e
.02
d.
0
mo2
.10
8
4
Analysis Of Spatial Relationships
250
500
Permeability, mD
Figure ll-7B.
..
0
1
2 3
Lag
4
5
6
Sample semivariogram for data with a step change.
Periodicity and Geological Structure
The effects of a periodic component upon the autocorrelation structure of a signal are
well-known (e.g., Schwarzacher, 1975, pp. 169-179). Suppose Z has a periodic
component with wavelength A-. These components are evident in the sample
semivariogram, )(h) (Fig. 11-8). The depth of the holes will depend upon the size of the
periodic component compared to any random component(s) present in Z. The depths of
the holes may also diminish with increasing lag, depending upon the nature of the
sedimentary system and the nature of its energy source (Schwarzacher, 1975,
pp. 268-274). In particular, since even strongly repetitive features in sediments are
usually not perfectly periodic, holes may be much shallower than the structure merits.
For example, a slightly aperiodic signal (Fig. 11-8 left) has a smaller first hole (Fig. 11-8
right) and much smaller subsequent holes than its periodic equivalent. Yet, the structure
represented by each signal is equivalent. So we must be careful using the
autocorrelogram or the semivariogram to detect cyclicity. If, however, evidence of
cyclicity appears in a sample plot, the cyclicity is probably a significant element in the
property variation.
279
Statistics for Petroleum Engineers and Geoscientists
2.0
1.5
1.5
1.0
1.0
0.5
0.5
f-A~
f-A~
f-A~
CIUaOOUOgiOU"'~~UOCI
0
10
20
Distance
Fig. ll-8.
30
40
0.0
0
10
20
30
40
Lag,h
Example permeability plots and semivariograms based on the series
Z(xi) = 1 + 0.4 sin(nxi/6) + ei, where ei - N(O, 1) and Zi (= 0, 1, ... )
is position, showing evidence of structure (cyclicity with A,= 12
units). The acyclic series is Z '(xi) = 1 + 0.4 sin(rrxi/6 + 5) + ej,
where 8 = O.Ol[xi/12] and [xi/12] is the greatest integer less than or
equal to zi/12. Left figure shows deterministic components of Z (x)
and z' (x). Adapted from Jensen et al. (1996).
Various nested, graded beds result in semivariograms that reflect underlying trends and
periodicity. More complex patterns may not be resolvable (especially real patterns!). See
the examples in Fig. 11-9.
280
Chap. 11
......
3
•••
.
.
5e'~Def!J
c.,!
~
"-l
._.
*
•
*
... .!!... .......It, ...... ~ ...... '" L
s
~
=
CIS
...
~
. .'I;.
*
*
)...
CIS
.:::;
LINEAR
s
<V
0
Q
500
Permeability, mD
""Q.
Ill"
it
d'
it
~
0
0
30
Lag
LINEAR
LOG-LINEAR
LOG-LINEAR
Permeability, mD
it
It
11!1>
VARLINEAR
V AR LOG-LINEAR
0
0
at
It
~
~
•• • • •
5
.
.,.
~
Q
~------·-----·
=
.
2000
N
. . .'lj
ClJ
Analysis Of Spatial Relationships
500
40
Permeability, mD
500
0
60
Lag
Figure 11-9.
Semivariograms for graded and nested patterns.
Statistics for Petroleum Engineers and Geoscientists
281
11-6 SEMIVARIOGRAMS IN REAL MEDIA
Geologic data can be much more complex than shown by certain idealized correlation
behavior. We show some examples in this section.
Autocorrelation in clastic formations tends to be scale-dependent, with different scales
being associated with laminae, bedforms, and formations. Additionally, autocorrelation
tends to be anisotropic; vertical and horizontal autocorrelation structure can be quite
different. Cylindrical anisotropy in the horizontal plane is also likely where the
deposition is unidirectional or current-generated (e.g., fluvial) in contrast to oscillatory,
wave-generated deposits that are more likely to be cylindrically isotropic (e.g., shoreface).
The statistical characterization of the properties of geologically recognizable forms is an
ongoing area of investigation. As an introduction, we illustrate some of these features in
the following items.
Deterministic Structure of Tidal Sediments
In a study of bed-thickness data for an estuarine Upper Carboniferous sequence in Eastern
Kentucky (Martino and Sanderson, 1993), autocorrelation analysis was used to determine
the spatial relationships of bed thickness. The correlogram shows distinct periodicity
with layer periods at 12 and 32. The first periodicity is interpreted to reflect spring-neap
tidal variations over 11 to 14 days and the longer range signal as having resulted from
monthly variations in tidal range. These results were supported by Fourier analysis.
The semivariogram can sometimes reveal "average" periodicities that are represented by
a significant reduction (to less than 50%) in variance at some lags. Two example
semivariograms from the Rannoch formation in two different wells, Fig. 11-10, show a
periodicity at 1.2 to 1.4 m (4 to 4.5 ft). This periodicity is similar to that clearly seen in
other intervals and is thought to be related to the (hummocky, cross-stratified) bedform
thickness.
282
Chap. 11
Analysis Of Spatial Relationships
2.
cu
c:
u
CIS
·;::
~
.......
ll)
u
1::
·~co
0·
Variance= 15831 mD2
-~
E
cu 0.
V'.l
0 20 40 60 80 100 120 140 160 180 200
Lag, em
Lag, m
Figure 11-10. Two examples of geologic periodicities in probe permeability data
from the Rannoch formation.
Statistics for Petroleum Engineers and Geoscientists
283
Semivariance/Variance
1.50
•
1.00
II
•, f .. 'n\-..."-
•,
•
0.50
•
(L1h "' 0.2 em)
2
0
3
4
(L1h"' 10 em)
(L1h"' 1 em)
0.00
0
10
20
30
40
50 0
200
100
Lag, em
•e
t
2.0cm
+
--,_.._
e
•
___t_
•
THIN BED
SCALE
---- ..... .....
--- --
t
8
••
~
•e
2.0cm
II
II
-r
150cm
•
G
•
••
INAPPROPRIATE
SAMPLE LENGTH
AND SPACING
I
~
BEDFORM
SCALE
Figure 11-11. Semivariograms at various scales in permeability data from the
Rannoch formation.
Determination of Nested Structure
There is a relationship between sample density, sample length, and the scale of geological
variation that one is trying to characterize. The larger-scale variation tends to dominate.
Several correlation lengths and scales of periodicity may exist in a formation, and each
Chap. 11
284
Analysis Of Spatial Relationships
requires a tailor-made sampling scheme. Figure 11-11 shows this. The general
dependency on scale coupled with the cyclic behavior makes it difficult to completely
characterize the autocorrelation behavior with even a fractal model.
Semivariograms and Anisotropy
Semivariograms in different directions can be used to demonstrate anisotropy. This could
be particularly useful where the trends are not so visually obvious as they are in
Fig. 11-12. The filled and open symbols in Fig. 11-12 are taken from data strings
perpendicular to each other.
1.50
Semivariance,
(mi./min.) 2
• Vertical
0 Lateral
1.25
•• •
1.00
•
1!111
1!111
«<J
1!111
1!111
0.75
•
0.50
0.25
0.00
0.0
•
0
0.5
1.0
0
1!111
•
1!111
0
1.5
2.0
2.5
3.0
Lag, em
Figure 11-12. Illustration of anisotropy in semivariograms from the
Rannach formation.
Clastic reservoirs are characterized by relatively weak autocorrelation in the vertical
direction and much stronger autocorrelation in the horizontal. In carbonates, the
preservation of bedding is not always so clear, but anisotropy may still be present.
285
Statistics for Petroleum Engineers and Geoscientists
11-7 BIAS AND PRECISION
With the tools now available to estimate autocorrelation, it becomes a natural question as
to which is preferable. As is usual, the preference of one measure over another depends
on the application, but we can also make some statistical judgments.
Yang and Lake (1988) investigated the bias and precision of the autocovariance and
semivariogram. They found that the autocovariance is in general negatively biased, that
is, it tends to underestimate the extent of autocorrelation. The source of this bias lies in
the estimate of the sample mean, which is required for the estimates of both Cov and p.
The sample mean does not appear in the definition for the semivariance, Eq. (11-6); it is
consequently unbiased. The semivariance is, therefore, the best tool to estimate
autocorrelation from this standpoint. These statements are necessarily oversimplifications;
the error behavior of the autocorrelogram depends not just on k and I, but also on the
underlying structure of the data. Priestley (1981, Sec. 5.3) discusses this and related
topics.
Both the autocovariance and the semi variance estimates are relatively imprecise beyond
one-half of the sample span, so that fitting models to experimental curves by regression
is discouraged. Since it is precisely the autocorrelation at large lags that is of interest in
calculating these measures, this can be a serious problem in practice.
The origin behind the lack of precision in the autocorrelation measures lies in the
successively smaller pairs available in the estimators, Eqs. (11-5) and (11-6), at large
lags. Caution is also required when interpreting estimates based on data sets with
significant proportions of missing measurements. In these circumstances, may be as
unreliable at some short lags as it is at very long lags. As a simple example, consider
the case of six measurement locations (/ = 6) with two data missing. If i = 2 and 4 are
missing, there is only one pair to estimate y(l) but two pairs to estimate y(2).
y
Li and Lake (1994) addressed these problems by an integral estimator that keeps
constant the number of pairs throughout the procedure. Maintaining the same number of
pairs in the calculation causes the semivariance estimate for the largest lag to be equal to
the sample variance. See the original reference for the estimator equation, which is
complicated. Figure 11-13 shows some of the results from using this moving-window
estimator.
Chap. 11
286
1.0
Analysis Of Spatial Relationships
I
I
;
0
I
•
i
I
i
.!
:;, rt''
/ i
Vertical (V)
. . . . . anum
0
•
aammaaaaa
New-1 (H)
New-2 (H)
Classical (H)
New-1 (V)
New-2 (V)
Classical (V)
i
I
i
m
Linear regression
Linear regression
Horizontal (H)
••
!
I
0.1
1
10
Lag distance, ft
100
1000
Figure 11-13. Semivariogram estimators for horizontal (H) and vertical (V)
data sets from the Algerita outcrop. From Li and Lake (1994).
The plot is on log-log scales to facilitate the interpretation of data with a fractal model.
But it is clear that the fluctuations present at large lags in the "classical" (Eq. (11-6))
estimator have been eliminated. The new estimator (New-2) performs better in this regard
than all the others. In addition to being more precise, the new semi variance estimator can
deal with multidimensional data (rather than along a line as has been the focus of the
discussion here) and does not require equally spaced data. The latter aspect means that the
new procedure is robust against missing data.
This "moving window" estimator may become an important standard method in the
future. Not only does it reduce the semivariance fluctuations at large lags, it also
smoothes out fluctuations at small lags, yielding semivariogram behavior that is easier to
fit to models near the origin. These effects are important for two reasons: (1) we are able
to interpret a model in difficult data situations (for example, with few wells or large well
spacing) and (2) because of the smooth near-origin behavior, we can use autofitting
methods, such as regression, to obtain a model. Being able to autofit semivariogram
functions would eliminate much of the subjectivity in current semivariogram analysis.
Statistics for Petroleum Engineers and Geoscientists
287
11-8 MODELS
Just as with PDF's and CDF's, we frequently have a need to report the extent of
autocorrelation in a fashion that is even more succinct than the estimates from data.
Moreover, some forward applications (see Chap. 12) require negative or positive semidefinite relations that the estimates may not provide. However, just as in the curve
fitting to PDF's, there is no strong physical significance to the models apart from the
association with the underlying geologic structure discussed above.
Finite Autocorrelation Models
We cover only the models most commonly used in reservoir characterization applications.
See Cressie (1991, Sec. 2.5) for others. All are, however, positive or negative semidefinite, as appropriate, and can be used in three-dimensional characterizations.
Exponential model of the autocorrelogram:
h
(11-7)
p(h) = exp(- A£)
This one-parameter model of autocorrelation is popular for its simplicity. The parameter
1\,£ is the correlation length. p(h) in this model is always positive and monotonically
decreasing. Consequently, it should not be used for cyclic or periodic structures. Clearly,
as A£ approaches zero, autocorrelation vanishes.
Spherical model of the semi variance:
for h
~
AR
(11-8)
for h > AR
This most popu!fr of all autocorrelation models is the three-parameter spherical model.
a 2 is the sill, a0 the nugget, and AR is the range. These quantities are illustrated in Fig.
11-14, which shows the spherical model fit to a small set of data.
288
Chap. 11
Analysis Of Spatial Relationships
Sill
Range
Nugget
0
0
Lag
Figure 11-14. Illustration of spherical semivariogram model.
The sill should approximate the sample variance, and the range the extent of the
autocorrelation. The nugget is an inferred quantity, since it can never be measured directly.
A semivariogram that has a nugget approximately equal to its sill exhits a pure nugget
effect and, similarly to having a small range, has no autocorrelation in the underlying
data. The pervasiveness of the spherical model has led to the incorporation of range, sill,
and nugget into the more general statistical terminology. For example, a data set with a
"large range" is strongly autocorrelated even though the spherical model may not apply at
all.
A semivariogram model that does include a degree of cyclicity and, therefore, has some
physical significance attached to its behavior is the hole effect model given by
The peaks and troughs (the holes) of this model are at solutions to the equation
h!A.H = tan(h!A.H)· The first peak is at h!A.H 3n/2 and the first hole is at h/A.H Sn/2.
The first hole has depth 0.13 of the nugget minus sill distance, 0'~ - 0'2 . The is the
deepest hole of any of the known models for 3-D correlation and is the minimum possible
for the model to be positive semidefinite. As seen earlier, deeper holes in some directions
are common in many sedimentary systems. Therefore, an anisotropic covariance is needed
to model this behavior with a model of the form
=
=
c
Statistics for Petroleum Engineers and Geoscientists
2
2
289
2
y(h) = a0 +(a - a 0) [1 - cos(h!?w)]
in the direction where the cyclicity is strongest.
Integral Scale
The correlation length and range are often used interchangeably, probably because they
both express the same general idea. However, they are not the same, each being unique to
the respective model. A quantity that also expresses the extent of autocorrelation is the
integral scale, defined as
00
(11-9)
AJ is defined for any autocorelation coefficient with a finite correlation, but, apart from
this, it is independent of the model chosen. Of course, AI = AL if the model is
exponential.
Infinite Autocorrelation Models
A growing body of literature suggets that many geological data sets do not have finite
correlation. Indeed, the data in Fig. 11-13 can be so interpreted. In this eventuality, there
is a class of models abstracted from fractal terminology.
The fractional Brownian motion lfBm) semivariogram,
y(h) =Yo h2 H
(11-10)
is a two-parameter model that contains a variance at a reference (h = 1) scale, y0 , and the
Hurst or intermittency coefficient, H. H takes on the role of the various lengths in
Eqs. (11-7) to (11-9), but, of course, y is unbounded in general. Figure 11-13 illustrates
one of the ways to estimate H by plotting yversus h on log-log coordinates. See Hewett
(1986) for discussion of other methods.
A related model is the fractional Gaussian noise (jGn), which has a much more
complicated model equation. However, Li and Lake (1995) have shown that, under most
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practical circumstances, fGn with 0 < H < 1 is equivalent to fBm for 1 < H < 2 as long
as H+l is used instead of H in Eq. (11-10).
The unbounded nature of y(and the simplicity of the model) makes fractal models
particularly appealing for geological characterization. But, it is conventional to truncate
the model at some upper and lower scales simply in order not to be extrapolating too far
outside the range of the data (Li and Lake, 1995).
Fitting Covariance Models to Estimates
The choice of appropriate autocovariance model based on data is often not straightforward.
A number of models might "almost" fit a sample semivariogram or correlogram.
Combinations of models can also be used, since the sum of covariance models will still
have the appropriate positive- or negative-semidefiniteness property. Carr (1995,
Chap. 6) and Journel and Huijbregts (1978, Chap. 4) give examples.
The fitting of a model must account for what behavior in the sample covariance plot is
"real" and what aspects arise from sampling variability. If the range or correlation length
can be related to a physical aspect of the system, then the model is better tied to the
geosystem being assessed. For example, the range has been observed to coincide with the
thickness of tidal channels (Jensen et al., 1994). Holes in the sample semivariogram
have been found to frequently reflect sedimentary cyclicities, and the nugget may represent
undersampled smaller-scale variability (Jensen et al., 1996). Therefore, an iterative
procedure may be needed where features observed in the sample covariance plot are
investigated for any relationship with sedimentary aspects. Another helpful test is to
apply a jack-knife procedure to the sample covariance to determine what behavior might
be unexplained by sampling variability. See Shafer and Varljen (1990) for an example of
this approach.
11-9 THE IMPACT OF AUTOCORRELATION UPON THE
INFORMATION CONTENT OF DATA
Until this chapter, we have ignored the fact that measurements made at nearby locations
in the reservoir may be correlated. All of the theory presented in Chap. 5 concerning
variability assessments of estimates implicitly assumes that all the data are independent
and makes no allowance for autocorrelation; a sample of size I is assumed to consist of I
uncorrelated measurements. We know, however, that this may not be correct. For
example, measurements within a facie may be highly autocorrelated, whereas
measurements from different facies may well be uncorrelated. How does correlation affect
the theory presented in previous chapters? Some idea about this was presented through
Example 2; we discuss the implications in the following paragraph.
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29"1
Autocorrelation reduces the information content of data. That is, I correlated
measurements will only be worth leff uncorrelated measurements, where left< I. The
greater the degree of correlation, the smaller is Ieffii. For example, we have for the
arithmetic average
if the I data are uncorrelated and come from a population with mean f.1 and variance a2 .
However, if the data have a first-lag correlation of p = p(l) ::f. 0, then (Bras and RodriquezIturbe, 1985)
{ a2 [
£ I p (1 - p) - p(l X- N Jl, I
1+ I
(1- p)2
Consequently, the variance of the estimate
pi)]}
X may increase.
Correlation of data can arise also when samples are bunched. For example, consider
the situation where a reservoir has been drilled as in Fig. 11-15.
Figure 11-15.
Schematic of variable sampling density for average porosities.
Would a representative porosity in the oil-saturated medium be given by (<h + ¢2 +
¢3 + ¢4)/4? Clearly, the answer is no; a simple average would probably be optimistic
since three of the four wells penetrate the largest part of the oil region. The ¢ values for
wells 1, 2, and 3 do not represent the same volume of material as does ¢4. A better
average porosity would be obtained by declustering the data-weighting each <P by the
volume it represents in the sum.
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Dealing with correlated data can be difficult. It is an area of continuing research for
which there are as yet more questions than answers. Barnes (1988) offers a recent and
helpful discussion of the problem. It is clear, however, that the first problem is to
become aware that the problem exists.
11-10 CONCLUDING REMARKS
The statistical study of autocorrelation is still in a state of relative infancy. We have
developed a fairly substantial number of tools to do it, some with a strong theoretical
base, but we are still learning how to use them. Of particular interest is how geologic
features can best be conveyed through the statistics. One thing seems obvious:
autocorrelation is very important to reservoir performance and we, therefore, are highly
motivated to make the connection between it and fluid flow. Chapter 12 describes a
means of doing this.
12
G
E L
I AL
Thus far, our discussion has been to develop and illustrate tools for the analysis of data,
primarily to characterize them using particular models and for diagnostic purposes. An
extraordinarily powerful use of statistics is to apply the models to construct synthetic data
sets for use in subsequent calculations.
This procedure was mentioned in Chap. 1, and we have implicitly used it several times
already. The idea is to use the data and any geological and physical information to
produce statistically plausible reproductions (realizations). This approach underlies
A
regression, where we take a model, Y = f(X, {3) + e, and data to establish the unknown
parameters ({3) of the model. Once determined, the model can be applied to produce
estimates of Y. These estimates will usually differ from the true values of Y but, if we
have modelled well, the errors will be small.
For regression, every time we set X = x0, the model will produce the same estimated
value for Y. Thus, regression is a method for producing identical realizations. Other
methods will produce different estimates each time the estimator is applied. Each
realization conforms to the statistical, geological, and physical information we
incorporate into the model. The different estimates, however, result because we have not
included sufficient information to uniquely predict the reservoir property at each desired
location in the domain.
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The general area of using statistical methods to create replicas of variable distributions
in the earth sciences has been called geostatistics. In this chapter we discuss the use of
statistical techniques and statistics to generate sets of data. We have already done this in a
simple way using the Monte Carlo technique for reserves estimates and stochastic shale
models in Chap. 3. Usually these sets will be on arrays or statistical fields that can be
used in a variety of calculations: reserve calculations (hydrocarbons in place), locating the
top of structures, or as numerical simulation input. Before becoming this specific,
however, we give a systematic summary of the basic techniques for generating statistical
fields based on only a limited amount of measured data, a problem that is among the most
basic in earth sciences.
12-1 STATISTICAL POINT OF VIEW
To begin with, we return to Fig. 3-15 to discuss a point of view that holds throughout
this chapter. The following remarks apply to any spatial dimensionality, but they are
easiest to visualize in two dimensions. Likewise, the remarks hold for all parameters
relevant to a given application, even though we discuss the figure as a univariate field.
The statistical point of view is that every point in the field (and each random variable)
has a probability density function (PDF). For measured points, the PDF consists of a
single value (a "spike" or zero-variance PDF, assuming no measurement error). For the
unmeasured points, the PDF has a nonzero variance and, without further restrictions, can
take on an arbitrary shape. For example, if we assume second-order stationarity, the PDF
must have the same first and second moments everywhere in the domain.
This point of view is central to the statistical techniques discussed in this chapter. But
we must remember that it is only a point of view; there is in reality a single value for
each variable at each point. If we sample one of the unknown points, assuming no
measurement error, it becomes known and its PDF converts to a spike (with a value, it is
presumed, that is consistent with the imagined PDF).
12-2 KRIGING
There are several statistical procedures for interpolating data. These procedures form the
basis for sophisticated techniques, so it is important that we understand the fundamental
ideas.
Kriging is a basic statistical estimation technique. Named after its originator, D.
Krige (1951), it was first used to estimate ore grade in gold mines. It was then
theoretically developed by researchers, primarily in France (e.g., Matheron, 1965). As we
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295
shall see, there are now many vanatwns of Kriging, but all start with the same
fundamentals and all are related to regression.
Consider a three-dimensional field as illustrated schematically (in two dimensions) in
Fig. 3-15. We seek an estimate of a property Z" at an unmeasured location (point o) that
is based on the Zi, i = 1, ... ,/known values.
I
z* = I~tizi
(12-1)
i=l
Equation (12-1) is a common interpolator. For the specific way we calculate the lti
described below, it is the Kriging estimator. It is linear in the Zi. The values lti,
i=l, ... ,I, are the Kriging weights. Clearly, if we determine the lti, we can calculate Z".
BlUE Estimators
Statistical estimation techniques are best linear unbiased estimators (BLUE). "Best," in a
statistical sense, means that the estimators have been arrived at through minimization of
some type of variance. "Linear" means that the estimator is a linear combination of data
values at known points, and, as we have discussed in Chap. 5, "unbiased" means that the
expectation of the estimator will return the true (or population) value. How these criteria
are imposed is evident in the following development of the Kriging estimator.
The linearity of Eq. (12-1) simplifies the procedure significantly. However, it also
means that the z* will be approximately normally distributed (Gaussian). Recall from
the discussion of the Central Limit Theorem in Chap. 4 that any random variable that is
the sum of a large number of independent, additive events will become Gaussian,
regardless of the underlying PDF of the events. If we view the ltiZi as the independent
events, the estimator in Eq. (12-1) satisfies these criteria iff is large. Unfortunately, z*
will be Gaussian even if the Zi are not; hence, statistical additivity (Chap. 5) requires us
to transform data (the ZD to normality before analysis and field generation. If the Zi are
normally distributed, the Kriging estimator will give the means of the imagined
distributions in Fig. 3-15, which are now themselves normal. A nice feature of the
normal PDF is that the z* will also be the most likely (the mode) value at the point o.
Chap. 12 Modeling Geological Media
296
Simple Kriging
To determine the weights A.i, we first define and then minimize the simple Kriging (SK)
.
2
vanance (YSK:
2
aSK=
E[(Z- Z*)2]
In this definition, z* is the estimate of Z at a single point. The Z itself is unknown,
but we can proceed without this knowledge. a}K is interpreted as the variance of the
distribution at the point being estimated (Fig. 3-15). Substitution of Eq. (12-1) and
rearrangement gives
(12-2)
where we have employed the convention that 0'~ = Var(Z), a;i = Cov(Z, Zi), and
ai~ = Cov(Zi, Zj) are known quantities. Written in this form, a}K is a function of
the weights, since the autocovariance model alone determines the autocovariances.
To minimize (jffK, we employ the now familiar argument (Chap. 9) that we seek a
point in Ai space such that da}K = 0. If the weights are independent, the minimal affK
will occur when
2
JaSK
~
.
= 0 for 1 =1,2, .. .,!
1
Performing the differentiations on Eq. (12-2) gives the following equations:
j =1, . .. ,I
which can be written in matrix form as
(12-3a)
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2
all
2
).,1
O"ol
=
2
all
AJ
(12-3b)
2
aoi
or in vector notation as
(12-3c)
The SK A-/s are therefore the solution to these normal equations. Equation (12-3b) is
similar to the regression matrix Eq. ( 10-7).
Strictly speaking, the requirement das2K = 0 locates only extrema in the Aj space, not
necessarily a minimum. We should, therefore, interrogate the second derivative of
to see whether we have arrived at a maximum or a minimum. This is left as an exercise
to the reader; however, the linearity of Eq. (12-3), caused originally by the linearity of the
Kriging estimator, means that the solution to these equations will always be a minimum
a}K
2
asK·
The elements in the above matrix and in the vector on the right side are
autocovariances. As will be seen below, these can all be obtained from the
semivariogram. The right side deals with autocorrelations among the known points and
the point o being estimated; the matrix elements account for autocorrelations among the
known points. The autocorrelation matrix contains variances along its diagonal and is
symmetric because of the reciprocal property of the autocovariance. The symmetry means
that the inversion of Eq. (12-3) is amenable to various specialized techniques. Although
the autocorrelation matrix contains nonnegative elements along its diagonal, it is not, in
general, positive definite, although this is frequently the case. Finally, if we invoke
weak stationarity (Chap. 11), all of the covariances become functions only of their
separation distances (or of the separation vector if the autocorrelation is anisotropic). The
locations of the Zi being fixed, the matrix of the
in Eq. (12-3b) need be inverted only
once to estimate Z: at other points.
a;]
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Chap. 12 Modeling Geological Media
We can eliminate the inner sum of the last term of Eq. (12-2) with Eq. (12-3a) to give
a short form for the minimized SK variance,
I
2
2 ~
2
aSK= (Jo- L...JA-iaoi
i=l
(l2-4a)
for the point o. Unlike the formula for the weights, this equation requires a variance of
the unknown Z. By convention, we evaluate
a~ from the sample variance or from the
semivariogram.
Many applications require estimates z1*, z2* , ... , Z1* on a grid of J values. The above
equations nicely generalize with matrix notation in this case. The Kriging estimator is
(12-5)
where z* is an M-element vector of the estimated values, Zan /-element vector of the
known values, and AT is an I x M matrix of the Kriging weights A-ij=
The subscripts on the Aij refer to the measurement position and estimation location,
respectively. Solving for the weights from Eq. (12-3c) and combining with Eq. (12-5)
gives an expression for the vector of estimates,
(12-6)
r,;
are the autocovariances between the known v~ues and
where the elements of r?and
between these values and the estimated points, respectively. Inverting :E can be
computationally intensive, if the number of data is large, since it is an I by I matrix.
But, as suggested above, the inversion need be performed only once for each point
estimated.
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299
Example 1 - Simple Kriging as an Exact Estimator. An estimator is exact if
it reproduces the sampled data points. Show that, under some fairly easy
conditions, the SK estimator is exact.
We begin with Eq. (12-3a) to estimate the j = kth known point:
Upon rearranging,
.
2
If the O"ik are independent of each other and not all zero, we must have
ltk - 1 = 0 and Aj =0
for j of:. k
Therefore, the weight for the kth point is one and all others are zero. As
would be expected, a~r 0 at this point from Eq. (12-4a). Since the point k
was chosen arbitrarily, this result applies to all the known points. Hence, the
Kriging estimator reproduces the values at the known points.
Exactness is a property not shared between Kriging and regression, and, in fact,
exactness will not apply here if the O"~ are dependent as might occur if the semivariogram
model used had a nonzero nugget. Indeed, you may show that the Kriging estimator
reduces to the inexact arithmetic average in the limit of the semivariogram model
approaching a pure nugget.
Kriging with Constraints
Many applications require additional constraints on the Kriging estimates, constraints that
arise from statistical grounds, from geology, from physical limits (nonnegative porosity,
for example), or from other data. There are several ways to constrain Kriging estimates,
but the one most consistent with the linearity of the Kriging estimator is the method of
Lagrange multipliers. We illustrate this method through an example. See Beveridge and
Schechter (1970, Chap. 7) for an exposition of Lagrange multipliers.
Chap. 12 Modeling Geological Media
300
Ordinary Kriging
Unfortunately, the z* from simple Kriging are biased (BLE, not BLUE), because we did
not constrain SK to estimate the true mean at unsampled locations. The SK bias can be
determined as
bz* = E(Z*) - E(Z)
=E
(~ l.; Z i )- E(Z) =~A; E(Z,) - E(Z) = (~(l.; - !) )E(Z)
Hence, the bias can be removed by forcing the A,i to sum to one. The problem of
determining the Kriging weights now evolves to a constrained minimization, that is,
minimizing
subject to the statistical constraint
a;K
I
I,t.i= 1
(12-7)
i=l
To apply the Lagrange multiplier technique to minimize the Kriging variance,
Eq. (12-2), subject to constraint Eq. (12-7), we form the following objective function:
The factor 2 before the last term is for subsequent mathematical simplicity. We can
either add or subtract the last term since it will be zero ultimately, but the form indicated
above has a slight mathematical advantage. J1 is the Lagrange multiplier, a new variable
in the problem.
Minimizing the objective function in the usual way (now setting dL
following equations for Ordinary Kriging (OK):
=0) leads to the
I
~
2
2
£,,,/•iO'ij = O'oj
i= 1
+ J1
which can be written in the following matrix form:
j = 1, ... ,/
(12-8)
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301
=
2
(Joi
1
The problem to be solved is nearly the same as for SK; there is an additional column
in the autocovariance matrix to account for the Lagrange multiplier and there is an
additional row to account for the constraint. Of course, there is now an additional
unknown, the Lagrange multiplier itself.
For large problems, say I> 10, there is a negligible difference in effect between
solving the SK equations and the OK equations, since the latter just involve a slightly
expanded matrix. However', the matrix, while still symmetrical in the form shown,
contains a zero on the diagonal that restricts the applicable inversion techniques. Just as
in SK, the matrix is invariant as the point to be estimated moves; hence, the
autocorrelation matrix need be inverted only once for multiple estimations.
We can combine the equation for the OK objective function and Eq. (12-8) to give the
minimized OK variance,
(12-4b)
or
Being constrained, the OK variance is usually greater than the SK variance because the
additional constraint does not allow the attainment of as deep a minimum.
Example 2 - Ordinary Kriging with Three Known Points. Consider the
arrangement of points shown below (Knutsen and Kim, 1970).
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Chap. 12 Modeling Geological Media
2
100
3
Figure 12-l.
Layout of known points (x) and point (o) to be estimated.
Estimate the value of z* at point o. The theoretical semivariogram is given
as
h)_ {
O.Olh
r( -
4
fl < 400
h > 4oo
Since the semivariance is written as a function of separation distance only, we
have assumed weak stationarity. Hence, the autocovariances and semivariances are related as (Chap. 11) Cov(h) = Cov(O)- y(h). We now calculate the
elements of the OK equations as
2
2
0"11 = a22
2
2
2
0"12 = 0"21 = 0"13 =
2
0"23
2
2
a 31
2
= 0"33 = Cov(O) = 4
=Cov(199) =Cov(O)- y(l99) = 2.01
2
=a 32
= Cov(20) = Cov(O) - y(20) = 3.8
2
0"0 1 = 0"0 2 =
2
0"0 3
= Cov(lOO) = Cov(O) - y(lOO) = 3
and the OK equations in matrix form are
4
2.01
2.01 2.01 1
A,l
3
3.8 1
A-2
3
4
2.01 3.8
1
1
4
1
A-3
1
0
J.1
=
3
1
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Solution of this set of equations yields A- 1 = 0.487, 4:2 = 0.256, A-3
and J.1 = 0.026, from which we take the estimate to be
= 0.256,
z: = (0.487)(10) + (0.256)(5) + (0.256)(15) = 9.99
where z1 = 10, q = 5, and z3 = 15. The OK variance can be obtained from
Eq. (12-4b) as u'fyK = 1.026. The z* estimate is nearly the same as the
arithmetic average, to which OK degenerates in the limit of no
autocorrelation.
As simple as this exercise is, it illustrates a number of points. First, note that
1\..2 + 1\..3 . This is because points 2 and 3 are so
close together that they are essentially acting as a single point in the estimate. If points
2 and 3 were spaced farther apart, A-2 would not equal .:t3. The sharing of weights for
clustered known points is known as a transfer of influence (Knutsen and Kim, 1970).
Fields with clustered points will have larger Kriging variances than fields with widely
separated points, a fact that obviously indicates the most efficient way to sample a
property in a field (but ignores scales of variation in geology and the financial aspects of
the sampling strategy).
.:t2 = .:t3 and that IL 1 is nearly equal to
Now suppose point 2 is moved so that it is directly behind point 3 from point o. We
would find, on solution, that 4:2 becomes negative and ~ exceeds one. (There is nothing
wrong with negative weights as long as the weights sum to one.) The negative weight
for 4:2 overcomes an undue influence from A-3 . By the same token, point 3 is shielding
point 2.
Another general property of OK is that if all the points, both known and to be
estimated, are so far apart from each other that they are uncorrelated, the Kriging estimator
becomes the arithmetic average. This property is consistent with the notion, first
discussed in Chap. 5, that the arithmetic average is the best estimator of the mean for a
set of uncorrelated data-as the points would certainly be in this case.
This discussion of moving sampled points around might lead one to believe that the
semivariogram and the data are independent. In reality, the semivariogram is estimated
from the data so that any shift in the sampling points will cause a change in the
semivariance. However, the change will be small if the semivariogram is based on a
large number of points. Such is presumed to be the case with the points in Example 2.
Note that it is possible to perform Kriging with fewer sampling points than are in the
entire set.
Chap. 12 Modeling Geological Media
304
Recall from Chap. 6 that the N 0 approach was described for obtaining sufficient
samples to allow the arithmetic average to be predicted to within 20% of the true mean
95% of the time. These Kriging results indicate that the No method is a simplification
and has to be modified to account for correlation among measurements. When the sample
locations are such that the measurements are correlated, more measurements need to be
taken to keep the variability of the estimated mean within 20%. See Chap. 11 and Barnes
(1988) for a discussion of the problem.
Finally, you may wonder why we resort to Lagrange multipliers to enforce the
unbiased condition in OK instead of renormalizing the SK weights by their sum. Such a
procedure does not result in a minimized a52K (the estimate would be LUE, not BLUE).
To see this, let f3 be the sum of the weights. We multiply and divide the second term in
Eq. (12-2) by f3 and the third term by {32. The minimization proceeds as before to result
in the following minimized SK variance:
I
2
2
asK= ao + (/3
2
~
2
- 2/3) .L..JAi a oi
i=l
This equation shows that a'ffK is the smallest only when f3 = 1, the same result as
Eq. (12-4a). Hence, such a renormalization will lead to a larger Kriging variance than
SK. This variance may or may not be larger than the corresponding OK variance, but the
Lagrange multiplier technique always leads to the deeper minimum.
Nonlinear Constraints
One of the more interesting, and largely unexplored, avenues in Kriging is the possibility
of adding more constraints. In the case of Kriging applied to permeability, in particular,
the linear unbiased constraint is weak because of the lack of additivity of this variable for
many situations (see Chap. 5). However, this can be modified somewhat using the
approach suggested inChap. 10 (e.g., Kriging log(Z) or z-1).
Well testing is a useful means of inferring the larger-scale flow properties of an aquifer
or reservoir. In this class of procedures, the permeability in the vicinity of a well is
inferred, usually by imposing changes in the rate of the well and measuring changes in
pressure. The permeability so obtained (we will call it kwt) is a good reflection of the
average flow properties of the reservoir, but it says little about its variability because it is
a global (that is, it applies to a large region) rather than a point measure. Moreover, it is
not clear how to average the point values within the test region to arrive at kwt·
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305
Campozana (1995) has added the nonlinear constraint kwt =-f(kl, k2, ... ,kM) to
Kriging through the Lagrange multiplier procedure. He deals with the global nature of
kwt by minimizing the sum of the Kriging variances at all points (recall that the Kriging
variances are all positive) and with the discrepancy in scales by allowing kwt to be related
to the point values through the following power-averaging formula:
The exponent m in this formula can be obtained through minimization, just like the
weights. Of course, the point-wise autocorrelation is maintained through the
autocovariances in the Kriging procedure.
Figure 12-2 shows the results of adding the kwt constraint to the permeability field.
The upper-left plot is a two-dimensional, fine-scale permeability field that is the test case
for this procedure. kwt for this field is known through a simulation of right-to-left flow.
The upper-right plot in Fig. 12-2 shows a reconstruction of the field using OK based on
data known at nine equally spaced points within the field; the lower-left plot shows the
result of the Kriging with the kwt constraint. The heterogeneity of the field is reproduced
much better with the kwt constraint. The example illustrates how Kriging can be used to
apply global information to improve arrays of point information.
To be sure, the improvement shown in Fig. 12-2 does not come without a price. The
nonlinearity of the constraint means that iteration is required to attain a minimized
solution and much larger matrices must be inverted. More subtly, the lack of additivity
may mean there is a basic inconsistency between the Kriging estimate and the power
average. However, the results of Fig. 12-2 are encouraging. Other nonlinear constraints
are possible, of course. See Barnes and You (1992) and Journel (1986) for an illustration
of how bounds may be added to Kriging.
Indicator Kriging
Kriging is a desirable estimator: it is linear, unbiased, and exact and has minimal
variance. Despite these attractive features, it has three other aspects that are less desirable.
1.
Kriging tends to generate a synthetic field that is too smooth-it underrepresents
the variability. This is a property that has been established from application;
however, it is also evident from the filtering aspects of the estimator itself.
Chap. 12 Modeling Geological Media
306
450
400
350
300
250
200
150
100
50
0
Figure 12-2.
Results of Kriging constrained to well-test data. Original field is in upper
left; ordinary Kriging reconstruction in upper right; simple Kriging with
well-test constraint is in lower left. The scale is in mD. From Campozana
(1996).
2.
Like regression, Kriging is deterministic; the same data values Zi will yield the
same estimates. Whether or not this is viewed as an advantage depends on the
application, but such a procedure cannot be used for uncertainty estimation.
3.
The fields generated are Gaussian. The restriction to the generation of Gaussian
fields (and the requirement of ensuring consistency) can also be bothersome.
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307
We can get around the deterministic nature of Kriging by regarding z* and cr~K (for
example) as the mean and variance of locally Gaussian PDF's (see below). Remarkably,
Kriging can be easily modified to eliminate all three of these problems by generating the
entire CDF of the estimates (Fig. 3-15) at each point. This procedure is called indicator
Kriging.
Before proceeding, you should review the material in Chap. 3 on CDF's. There we
discovered that a CDF consists of a plot of the frequency of occurrence F(z) of a random
variable Z less than some limit z. F(z) was nondecreasing and consisted of two types:
continuous CDF's, appropriate for continuous variables, and discrete CDF's, appropriate
for discrete variables. Recalling the definition of probability (Chap. 2), for a given cutoff
value zc, F(zc) represents
F(zc) =
Number of samples with Z ~ zc
Total number of samples
Other terms for zc are the critical or threshold value. zc varies between a minimum
zmin and a maximum zmax and can take on several values.
Let I(zcn) for n = 1,.. .,N represent an indicator variable defined as
(12-9)
I(zc ) is an on-off (Boolean) variable because it can take on only two values. N is less
n
than or equal to the number of data values. The estimated CDF follows from
1
N
F(zc) = N '"JJCzc)
n=l
and letting zc run from Zmin to Zmax· p* (zc) is the sample or empirical CDF of z*.
We form the indicator semivariogram Y[ in exactly the same fashion as the
semivariograms discussed in Chap. 11. These devices have the same meaning as they did
there except they are now expressing the continuity (extent of autocorrelation) of z*
values less than Zc· It is clear from this comment that the indicator semivariogram YJ can
be used to concentrate on a particular range of the Z that is known to be most significant
308
Chap. 12 Modeling Geological Media
to the application, e.g., the connected large-permeability range in flow simulations. 11•
of course, is always less than one because 0 5:./(zcn) 5:. 1 since /(zen) is either zero or
one.
Perhaps the most desirable feature of the indicator semivariogram is that it is robustthat is, YI does not depend strongly on the nature of the distribution of the Zi nor is it
affected by the presence of extreme values. The latter factor makes the indicator
semivariogram very useful in assessing autocorrelation in noisy data.
It is also clear that 'YJ will depend on the value of zc chosen, a decided increase in
complexity compared to previous semivariogram analyses. However, the semivariogram
is frequently robust to the zc, in which case a single 'Yt is satisfactory for the subsequent
construction. If zc is the median value in a data set, the subsequent Kriging is known as
median indicator Kriging.
The actual indicator Kriging proceeds exactly as in Kriging, except YJ is used in place
of rand the data values Zi are replaced by their respective indicators /i· (Replacing the
data values with indicators and using the indicator semivariogram is a form of
nonparametric statistics.) The following example indicates how to use them.
Example 3- Indicator Kriging for Three Known Points. Rework Example 2
to generate a CDF forT at the point o. Take the indicator semivariogram to
be
.
r(h) --
{ h/1600
o.2s
h < 400
h > 400
and independent of the value of J(zc n ) chosen. As before, we use this to form
the Kriging equations.
0.25 0.1250.1251]
0.125
0.25 0.238 1
[
0.125 0.238 0.25 1
1
1
1
0
0.188
0.188
1
where we have now used Cov(h) =Cov(O) - yJ(h). The solution to these
equations yields the same weights as before: /\. 1 =0.487 and 10, =/\.3 =0.256.
Statistics for Petroleum Engineers and Geoscientists
309
rr
Since is independent of zc, these weights will remain the same throughout
the example. For zcl = 5, the indicators for points 1, 2, and 3 are 0, 1, and
0, respectively, from which we calculate
F*(5) = 0.487(0).+ 0.256(1) + 0.256(0) = 0.256
for the CDF value corresponding to zc1 = 5. Similarly, for zc2 = 10 we have
I= (1, 1,0} andforzc3 = 15,/= {1, 1, 1}. ThesegiveF*(10)=0.743 and
F*(15) = 1, which give the discrete CDF at the point o (Fig. 12-3). To
estimate uncertainty, we sample the CDF in the same fashion as was done in
earlier chapters.
The discreteness of the F* depends on the number of known points. In Example 3, the
CDF has only three steps because there are only three points. Clearly, the more data
there are, the better is the approximation to the population CDF. As in the other types
of Kriging, the autocovariance matrix does not change from point to point (if r1 is truly
independent of zc)· Even so, generating F* can become time-consuming if the full CDF
must be constructed at every point in a field that has a large amount of data. But, unlike
ordinary Kriging, the weights A.i here must be nonnegative, otherwise the properties of
the F* (Chap. 3) will not, in general, be satisfied.
Finally, and most interestingly, the indicator variable may be associated with some
geologic feature (e.g., I= 0 is shale, I= 1 is sand). When this occurs, indicator Kriging
is estimating the probability that the particular feature exists at the point of estimation.
Such a procedure, expanded to include several different types of indicators, offers the most
direct way to convert geologic classifications into a synthetic field. We discuss this
further in Section 12-4.
When applying indicator Kriging, users will typically choose from one to five cutoffs
based on geologic or engineering criteria, e.g., net pay, lithologies indicated from wireline
logs, flow units > 100 mD, etc. In this fashion, indicator semivariograms will have
geologic and/or engineering significance. However, such few cutoffs will generate a local
CDF that is segmented and will not fully represent the range and shape of the actual
CDF. Such CDF's will give a poor representation of the local statistics; the median in
Fig. 12-3, for example, is estimated to be about 7, whereas it is actually 10 from the data
and from the nonsegmented CDF's.
Chap. 12 Modeling Geological Media
310
1.0
-
Segmented
/
representation~
0.8
ofCDF
-
/
/
/
/
I
I
0.6-
'\
I
b
F"' or F
0.4 -
,'
I
I
Data
t>
/
0.2 -
/
/
Estimate
"7
/
Median
/
0
I
0
10
5
15
z
Figure 12-3.
Discrete CDF of Z at the point o (Fig. 12-1) contrasted with the
CDF of the Zi from Example 3. The indicator CDF is different from
the empirical CDF (values between 5 and 10 are less probable)
because it is now conditioned on the data.
Figure 12-4a shows that segmentation produces artifacts in regions of the CDF where
the curvature is large, where the estimated CDF "short hops" across the empirical CDF.
Short hopping can be corrected by selecting additional cutoffs (increase the number of data
or categories) in the large-curvature areas (Fig. 12-4b).
Additional semivariogram modeling is required for these nongeological/engineering
cutoffs, but usually only one to three additional cutoff points are needed.
12~3
OTHER KRIGING TYPES
This discussion has far from exhausted the different possibilities of Kriging. Other forms
are briefly described below.
Statistics for Petroleum Engineers and Geoscientists
311
Universal Kriging
This form seeks to account for drift or a long-range autocorrelation in the data by
subjecting the data to a preprocessing step. The preprocessing can be as basic as fitting a
low-order polynomial to the original data or more sophisticated, such as segmenting the
data according to the geologic patterns.
1
F* or F
1
a
F"' or F
b
Empirical
CDF
Empirical
o .~.----:;;.,...---------o
Zmin
Zcl
Zc2
z
Figure 12-4.
CDF
Zmax
z
Illustration of the effect of adding additional cutoff points to eliminate
"short hopping."
Disjunctive Kriging
This is another way to deal with the Gaussian nature of Kriged data. Here the data are
first rendered Gaussian by the appropriate transform before Kriging proceeds.
The ideas behind universal (e.g., stratigraphic coordinates) and disjunctive (e.g., the
p-normal transform) Kriging have become common practice. The remaining form of
Kriging is more involved.
312
Chap. 12 Modeling Geological Media
One of the most useful generalizations is to define the original Kriging estimator,
Eq. (12-1), in terms of more than one data set. For example, if the need is to estimate
z* based on known Zi and Yi, as might be the case where we would estimate
permeability from permeability and porosity data, the estimator would be
I
I
i= 1
i=l
z*-~
lt·Z·+ ~ll
~v·Y·
-~ll
The weights, vi, account for any change of units between Y and Z. The ensuing Kriging
manipulations are exactly the same as those above, but with an expanded set of equations
to be solved since terms involving the correlation between the Zi and the Yi will appear.
Co-Kriging is a powerful tool that can incorporate a broad range of reservoir data in its
most general form.
12=4 CONDITIONAL SIMULATION
As we mentioned above, Kriging fields are deterministic and cannot be used to quantify
uncertainty. In the following, we review some of the more common procedures,
collectively termed conditional simulation (CS), to generate a stochastic random field.
The term conditional arises because, as we shall see, the fields so generated are exact, just
as with Kriging. All techniques, which are more algorithms than mathematical
developments, use some form of Kriging. In all cases we are to generate a random field
consisting of values inj =1, ... , J cells based on i =l, .. . ,I sampled data points. There
are therefore (J -1) points to be estimated or simulated initially based on I conditioning
points. We have previously determined the appropriate semivariogram model either from
the I data points or from geology and we assume the values at the conditioning points are
free of error.
Sequential Gaussian Simulation
Sequential Gaussian Simulation (SGS) is a procedure that uses the Kriging mean variance
to generate a Gaussian field. The entire procedure is the following:
1.
Transform the sampled data to be Gaussian. The most common technique is the
normal scores technique (Chap. 4). This is a natural precursor to any Gaussian
technique.
Statistics for Petroleum Engineers and Geoscientists
2.
3.
4.
5.
6.
7.
313
Assign each of the (J - I) unconditioned cell values to be equal to those at the
nearest conditioned cells. The values in the conditioned cells do not change in
the following.
Define a random path through the field such that each unconditioned cell is
visited once and only once.
For each cell along the random path, locate a prespecified number of surrounding
conditioning data. This local neighborhood, which may contain data from
previously simulated cells, is selected to roughly conform to the ellipse range on
the semivariogram model.
Perform ordinary Kriging at the cell using data in the local neighborhood as
conditioning points. This determines the mean of the Gaussian distribution z*
and the variance crJ K· The local CDF is now known since the mean and
variance completely determine a Gaussian distribution.
Draw a random number in the interval [0, 1] from a uniform distribution. Use
this value to sample the Gaussian distribution in step 5. The corresponding
transformed value is the simulated value at that cell.
Add the newly simulated value to the set of "known" data, increment I by 1, and
proceed to the next cell as in step 4.
Repeat steps 4 to 7 until all cells have been visited. This constitutes a single
realization of the procedure. Multiple realizations are effected by repeating steps 3 to 7.
All the values in all realizations are back-transformed to their original distribution before
subsequent use.
At each step in the random path through the grid, a new conditioning data point is
added to the list of original data points. Use of the local neighborhood is to avert the
problem of inverting an ever-increasing Kriging matrix (Eq. (12-6)) as more and more
points are estimated. It is also possible to use a prespecified maximum number of nearest
conditioning data, including previously simulated values, in the estimate. In this way the
Kriging matrix will never be more than this number of points within the ellipse and the
SGS method will be able to handle large fields. Of course, ordinary Kriging in this
procedure can be replaced by simple Kriging or co-Kriging as the need dictates.
The local neighborhood restriction, however, can not preserve large-scale trends. This
can be modified to some extent by increasing the local neighborhood at the expense of
greater effort. Obviously, some care is required in using fractal semivariograms.
Alternatively, we can select a set of conditioning points that are a mix of points inside
and outside of the neighborhood. The estimated points in SGS act as "seed points" for
new stochastic features (e.g., low-, medium-, or high-value regions) in the random field.
Since the SGS procedure ensures continuity in the random field according to the
semivariogram, these stochastic features can persist and win be geologically acceptable if
the semivariogram is consistent with the geologic structure. Cells far from conditioning
Chap. 12 Modeling Geological Media
314
points can take on values that are equiprobable over the full range of the empirical CDF;
hence, SGS maximizes uncertainty in this sense.
Random Residual Additions
In the following, we discuss a procedure that is computationally more intensive but does
not involve the subjectivity of determining a local neighborhood. We call it the
procedure of random residual additions (RRA).
Figure 12-5 illustrates the procedure; the data points are indicated with x's, and we are
to fill in a random surface in this one-dimensional example. Five steps are required to
produce the desired field.
z
0
Residual
Distance
Figure 12-5.
1.
Schematic of random residual addition procedure. z0* K is ~he first Kriging
surface (deterministic); Zs is the unconditioned surface; ZsK is the second
Kriging surface; Z 0 is the final surface.
Generate a Kriging surface. This surface, shown as the dashed line z0* K in
Fig. 12-5, has the usual attributes of Kriging, the most important here being the
exactness and the determinism.
Statistics for Petroleum Engineers and Geoscientists
315
2.
Generate an unconditioned surface. This surface, shown as the erratic solid line Zs
in Fig. 12-5, has the desired statistical properties of the final surface, the
appropriate second moments as conveyed through the semivariogram. But, of
course, being unconditioned, this surface does not pass through the measured data.
We discuss ways to generate this surface later. In general, many of the statistical
tools used in this generation are the same tools used in the Kriging step. The
unconditioned surface is not deterministic.
3.
Generate a second Kriging surface. This step is the most important of all. The
* in Fig. 12-5, passes through the
second Kriging surface, shown as ZsK
unconditioned surface at spatial coordinates that coincide with those of the
* surface, therefore, treats i =1,2, .. .,I points
measured data in Fig . 12-5. The ZsK
on the Zs surface as known.
4.
Subtract the unconditioned surface from the second Kriged surface. These
differences are known as residuals. The residuals are zero at the measured data
locations because of the coordinate conditioning in step 3.
5.
Add the residual surface to the original Kriged surface (step 1). Because the
residuals are zero at the measured points, the original data will be reproduced in the
synthetic surface.
The final surface Z 0 in Fig. 12-5 has many desirable properties: it matches the
measured data, has the appropriate degree of heterogeneity (if there is consistency between
the statistical devices used in generating the various surfaces), is unbiased, and remains
minimal. Many of these attributes, carry through from the ordinary Kriging step because
of the linearity of the operations.. Most importantly, the final surface is stochastic; steps
2 to 5 will be different for each execution and constitute a realization for the entire
process.
Throughout this book we have been making distinctions between deterministic and
random processes (although both may be treated statistically). The RRA procedure offers
a clear demonstration of the combination of the two (i.e ..,. it is a hybrid method). The
determinism comes from step 1 and the randomness. from steps 2 to 5. Indeed, many
applications may be so deterministic that only the ordinary Kriging step is necessary. Of
course, other surfaces may be so erratic that the determinism is negligible, but these
limits may not be so apparent in the original data. R'RA, then, offers a way of blending
these two aspects in, it is hoped, the appropriate amounts.
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Chap. 12 Modeling Geological Media
Generating an Unconditioned Surface
There are two basic techniques for generating an unconditioned surface: spectral methods
and averaging methods.
Spectral methods rely on the ability to represent a synthetic field with a Fourier series
(Shinozuka and Jan, 1972). The key information here is a plot of the amplitude of each
term in that series versus its frequency, known as a spectral density plot, which can be
analytically related to autocovariance functions (Jenkins and Watts, 1968). This means
that spectral methods are slightly limited in their ability to use a large number of
autocovariance functions, especially those related to fractal semivariograms (Bmining,
1991). Spectral methods are difficult to implement in more than one dimension;
however, a version of spectral methods, the turning-bands method (Montoglou and
Wilson, 1985), is quite efficient This method relies on projecting a one-dimensional data
set throughout a two-dimensional field by a series of geometrical constructions around a
rotating line. The price for this efficiency is a slight, spurious periodicity in the
generated field (Ghori et al., 1990).
Averaging methods exhibit more flexibility than spectral methods. They range from
the relatively simple source-point method (Heller, 1972) to the most complicated matrixdecomposition method. The source-point method is simple because it does not require a
matrix inversion. It can generate a wide variety of autocovariance functions, but fields
with a prespecified function must be developed by trial and error (Bhosale, 1992).
The matrix-decomposition method (MDM) is the most computationally intensive of
the averaging methods (Yang, 1990), and its advantages are to be balanced against the
computation cost. The intensiveness results from the repeated inversion of an
autocovariance matrix. Even though the many inversion procedures are quite efficient,
this task is still challenging even on the most advanced computers. Various
approximations and the use of parallel computing seem to be of great advantage here
(de Lima, 1995).
Both the foregoing CS procedures generate Gaussian fields that are frequently at odds
with actual (geologic) distribution of properties. For example, the procedure can generate
a stochastic feature that a geologist, based on sedimentary principles, knows is
deterministic. These difficulties can be avoided with a careful geologic study that precedes
or, better yet, is done in conjunction with the field generation. With this in mind, CS
can be done entirely within the geologic framework or the geology can be superimposed
through a multilevel CS. Chapter 13 shows some examples. The principal difficulty
with these CS procedures is that they tend to be time-consuming, especially if multiple
realizations are desired. Of course, being able to generate multiple realizations is the
principle justification for CS. However, we can use indicator mathematics to generate
geologically based realizations, as the following paragraphs describe.
Statistics for Petroleum Engineers and Geoscientists
317
Generating Facies Distribution
This method is called the sequential indicator simulation-probability distribution function
(SIS-PDF) method. The objective is to generate a random field that classifies each cell
into a facies category. Once each cell is so assigned, we can associate a property to the
cell from the PDF of that facie.
We begin by assuming that the spatial distribution of facies categories is mutually
exclusive, that is, two facies cannot exist at the same cell. The spatial pattern of each
facie is governed only by its overall probability of occurrence and its semivariogram (e.g.,
structural type, range, orientation, and anisotropy). The overall facies PDF represents the
field-wide occurrence of each category; it is usually obtained from well data or a facies
map. Indicator simulation is ideal for modeling such "discrete" variables as these
(Deutsch and Journel, 1992; Alabert and Massonnat, 1990). The SIS-PDF procedure for
categorical indicators has the following steps (Goggin et al., 1995).
1.
Define a random path through the reservoir model such that each unconditioned
cell is visited once and only once. Conditioned cells contain wells that are
initially given known facies categorical values.
2.
For each cell along the random path, locate a prespecified number of conditioning
facies data, including facies data from wells, pseudowells, and previously
simulated cells.
3.
Using indicator Kriging (above), estimate the conditional probability for each
facies category. Each facie will be estimated independently using individual facies
semivariograms.
4.
Normalize each facies probability by the sum of probabilities over all facies and
build a corresponding local conditional CDF from the normalized probabilities.
5. Draw a uniform random number in [0, 1] and determine the simulated facies
category in the current cell by sampling the local CDF generated in step 4.
6.
Repeat steps 2 to 5 for each cell in the random path, paying attention to the local
availability of conditioning facies data in the previously simulated cells. New
facies "realizations" are obtained by reinitializing the random path through the
model, beginning at step 1.
Goggin et al. (1995) used this procedure for modeling facies patterns and their
uncertainties in sand-rich turbidite systems. Facies categories, identified from interpreted
Chap. 12 Modeling Geological Media
318
seismic-amplitude map extractions, calibrated using core and log data, are (a) channels,
(b) overbank deposits, (c) turbidites, and (d) distal lobes. Major facies trends were
preserved by sampling the interpreted facies maps in "high probability" areas (e.g.,
pseudowell picks). Therefore, the SIS-PDF technique can model local facies uncertainty
at the scale of cells in a full-field flow simulation.
·12-~
SIMULATED ANNEALING
Simulated annealing (SA) is a method for producing a field with known properties. The
procedures attempt to simulate physical processes, such as melting, freezing, and genetic
reproduction, through a variety of algorithms. By the same token, the number of
applications has also been diverse. Its use in reservoir characterization is fairly recent
(Farmer, 1989) although it shows great promise (Datta Gupta, 1992; Panda and Lake,
1993).
Simulated annealing overcomes many of the difficulties with the methods described
above. The data do not have to be normally distributed (in fact, any type of distribution
is acceptable), stationarity is not necessary, conditioning is easy, and there are virtually
no limits on the type of autocorrelation or heterogeneity a given field may possess.
Indeed, SA can entirely ignore autocorrelation if necessary. The single largest advantage
of SA is that it can incorporate several constraints, each from a different source, as long
as these do not conflict. Hence, it is possible to generate a field that, in principle,
satisfies all of the data gathered in a specific application. Finally, SA, as a .class of
techniques, is very simple and direct. Balanced against these advantages are the facts that
SA tends to be quite computationally intensive and it has an uncertain theoretical
pedigree. The trade-offs among these items should become clear as we discuss the main
SA techniques, but first we cover some background items.
·
Background
If a metal or a glass is heated to near its melting point and then cooled to a given
temperature T, the properties of the material depend on the rate of the cooling. For
example, the material might be stronger, less brittle, or more transparent when cooled at a
slower rate. The explanation for the different properties is usually based on diffusion
arguments; if cooling is too fast, there is not enough time for physical defects or
entrained gas to diffuse from the material. The process of slow cooling is called
annealing.
From a thermodynamic viewpoint, each step in the cooling results in an equilibrium
state or a minimum in the free energy surface. This is idealization because true
equilibrium at any temperature Tis attained only for an infinitely slow cooling rate. The
Statistics for Petroleum Engineers and Geoscientists
319
end of each cooling step, then, must represent a local equilibrium or a local mtnimum in
the free energy surface.
A central tool in the application of SA is the Boltzman or Gibbs cumulative density
function given by
F(&) = exp(-Llli/D
(12-10)
where AE is the change in the energy, Tis the temperature, and F is the frequency of
occurrence of a given AE. Both T and the energy change are positive. The CDF in
Eq. (12-10) is the complement of the one-parameter exponential distribution function
first discussed in Chap. 4. Based on this equation, we see that the most likely energy
change at a given T is zero and that nonzero energy changes become less likely as T
decreases.
It is important to remember that SA is essentially an argument by analogy; the
quantities we have been calling energy and temperature are really just thermodynamic
names for variables that behave like them. The energy, in fact, plays the role of the
objective function in previous work. A typical form for an objective function is
M
E = I,mm (zm- zcm)2
m=l
where there are m =1,2, ... ,M constraints, and mm is a weighting factor, selected in part
to make the units consistent in this equation. The z and zc are actual and computed
statistics of the field, respectvely; usually they are a function of the Zi, i =1,2, .. . ,I
points to be generated. This equation is desirably vague for it is here that all constraints
on a given field apply. The z may be anything; we will show examples below of z as
points on a semivariogram, global averages, and tracer data. The key property of E is that
it be nonnegative, but this can be satisfied by other forms. For example,
M
E= I,mm lzm-zcml
m=l
also satisfies the nonnegative requirement. As we shall see, we take only differences of E
in the SA procedures, rather than derivatives as in conditional simulation; hence, the
absolute value form is as convenient as the squared form.
Simulated annealing can be implemented in several ways. We cover two basic types:
the Metropolis and the heat-bath algorithms. As is usual, we generate a grid ofj=l,2, .. .,J
Chap. 12 Modeling Geological Media
320
properties Zj that satisfy the constraints and that are also drawn from a specific empirical
(experimental) CDF.
Metropolis Algorithm
This algorithm, first proposed by Metropolis et al. (1953), is the simplest SA algorithm.
It has the following steps.
1.
Generate a random field on the grid of J values by drawing some initial Zj from
the desired empirical CDF.
2.
Based on these values, calculate m for theM constraints.
3.
Calculate the objective function Eotd· If this is the first outer iteration, set the
initial temperature to E 0 zd·
4.
Choose a gridblock randomly and draw a new value
z
z:;ew for this cell from the
empirical distribution.
5.
Recalculate the objective function Enew-
6.
If Enew is less than Eold• accept the new value (that is, replace the original Zj
with z:;ew), set E 0 zd = Enew• and return to step 4.
7.
If E new is greater than E old• calculate ,1£ = E new - E old and the Gibbs
probability F from Eq. (12-10).
8.
Pick a random number r from a uniform distribution between 0 and 1. If r > F,
reject the
9.
z:;ew and return to step 4.
Otherwise, go to step 9.
If r < F, accept the new value (even though t.E has increased), let E 0 zd = Enew,
and return to step 4.
Repeat steps 4 to 9 until a suitable number of acceptances has occurred. These
steps constitute the search for a local minimum at a given temperature. Usually
the number of accepts should be somewhat greater than J, but it is not necessary
to be more specific since we will ultimately be seeking a global minimum after
the temperature has been lowered.
10. Test for convergence. Check to see if !ill is below some preset tolerance. If it is,
terminate the procedure; if not, continue to the next step.
Statistics for Petroleum Engineers and Geoscientists
321
11. Lower the temperature according to some cooling schedule such as T new =rf Told
and return to step 4. rfis a reduction factor (rt< 1) that is typically around 0.8.
The Metropolis procedure involves two levels of cycling; the inner level, steps 4 to 9,
at constant temperature, is noniterative and the outer level, steps 4 to 10, is iterative.
Figure 12-6, a plot of E versus the cumulative cycling counts of three outer cycles,
illustrates how convergence is attained.
In this figure, T is decreasing to the right with inflections intended to show when a
new temperature is adopted. E decreases at each fixed temperature, but not necessarily
monotonically; the cycle for T1 illustrates a local minimum forE there. Steps 7 to 9 are
intended to boost E over such local extrema. Since the algorithm uses E at two levels
only, it is a form of a Markov process.
The generality of this procedure is somewhat offset by the ad hoc nature of many of
the steps, for example, the number of inner cycles and the rate of cooling. To set these
values for a particular application normally requires some trial and error. From a rigorous
standpoint, the most serious problem with SA is that there is no guarantee that the
process will converge. However, the procedure is quite flexible, as the following simple
example shows.
E
1_
Tolerance
T ----------------------lnteration Number
Figure 12-6.
Schematic convergence of simulated annealing.
322
Chap. 12 Modeling Geological Media
Example 4 - Simulated Annealing with the Metropolis Algorithm (from
M.N. Panda, personal communication, 1994). Use the Metropolis algorithm
to generate a four-cell (J = 4) grid with a geometric mean z = 340. Use the
absolute-value form of the energy definition.
For later comparison, let the true distribution be
where the subscripts indicate the cell number. Of course, we know only the
value of the geometric mean before we do the annealing.
In this case there is only M = 1 constraint and we perform the inner cycle
seven times at each temperature. The discrete experimental PDF is the set of
nine equally likely values (100, 200, 300, 400, 500, 600, 700, 800, 900).
This means that if a random number falls between 0 and 0.111, the selected
value is 100; between 0.444 and 0.555, it is 500; etc. Table 12-1 shows the
steps for the results for the first temperature T0 = 106.
Table 12-1. Inner cycle for Metropolis example.
2
3
4
5
6
7
Move Rand. Cell Cell Cell Cell Geo.
Cell
4
Mean
No.
1
2
3
Initial 400
1
1
3
4
2
4
3
0.82
0.59
0.99
0.96
0.01
0.61
0.13
800
600
600
600
600
600
600
500
500
500
500
500
100
100
100
400
400
400
900
400
400
400
200
500
500
500
500
900
900
500
500
447
532
495
606
573
383
331
278
8
E
106
191
154
265
232
42
10
63
9
Delta
E
85
48
111
78
-190
-33
53
10
11
12
Gibbs Rand. DeciPro b. No. sion
0.45
0.64
0.35
0.48
0.61
0.57
0.51
0.78
0.44
0.38
Reject
Accept
Reject
Accept
Accept
Accept
Accept
The first column is the randomly selected cell, and the second is the random number
Column 7 shows the current geometric mean.
for selecting the trial value
Columns 8 and 9 show the energy (E) and the energy change (Llli), respectively. The
Gibbs probability and the corresponding random number are not needed unless the
z?ew.
323
Statistics for Petroleum Engineers and Geoscientists
energy change is positive. Three trials were accepted even though the energy change
was positive. The energy has decreased from 106 to 63 (not monotonically) during
this cycle.
Table 12-2 further illustrates the entire process by showing only the accept steps
until convergence.
Table 12-2. Acceptance steps for Metropolis example.
Ternn.
106
85
Move Rand. Cell
Cell No.
1
Cell
Cell
Cell
2
3
4
Geomean
0.59
0.96
0.01
0.61
0.13
0.06
0.66
0.35
0.18
0.09
0.11
0.65
0.58
0.63
0.58
0.47
0.31
0.44
500
500
100
100
100
100
500
500
500
500
500
500
500
500
500
500
500
500
400
400
400
400
200
200
200
200
200
200
200
200
600
600
600
500
500
300
500
900
900
500
500
500
500
400
400
100
100
100
100
100
100
100
100
100
495
573
383
331
278
278
416
394
299
211
211
266
350
350
350
334
387
341
1
4
2
4
3
3
2
4
1
4
3
1
68
54
3
1
3
3
1
3
600
600
600
600
600
600
600
600
200
200
200
500
500
500
500
500
900
900
Trial
E
154
232
42
10
63
63
75
53
42
129
129
75
9
9
9
6
46
0
Delta
Gibbs
E
Pro b.
Rand.
No.
48
78
-190
-33
53
0
13
-23
0.64
0.48
0.51
0.44
0.61
0.38
0.86
0.53
0.36
0.18
0.48
0.21
-11
88
0
-54
-66
0
0
-3
40
-6
The horizontal lines show when the temperature has been lowered (we have
used rt= 0.8 here). Again, several trials were accepted even though the
energy increased; indeed, three trials resulted in no decrease of the energy
whatsoever. We can illustrate the nature of the convergence in this simple
problem with Fig. 12-7.
324
Chap. 12 Modeling Geological Media
As illustrated, the convergence is not monotonic because of the possibility of
accepting a _positive energy change. The final answer,
is not the same as the actual one. The converged solution, which may be
regarded as a realization of the process, differs from the true distribution that
we gave at the first of the example because the objective function contained
no information regarding the spatial arrangement of the values. However, the
final solution has the desired geometric mean.
Tz .....,.j--T3 . . . . . I......_T4
250
200
>.
150-
C)
.._
(!)
c
w
100
so0
5
10
15
Count
Figure 12-7. Convergence rate for Example 4.
20
Statistics for Petroleum Engineers and Geoscientists
325
This example illustrates both the strength and weaknesses of SA. The objective
function is statistical but it does not deal with spatial arrangement, although this could
have been included. The procedure converges, but it converges slowly, especially toward
the end of the process where most of the Gibbs trials are rejects. The multiple rejections
can significantly increase the computation effort of the process.
Heat Bath Algorithm
The second SA algorithm eliminates the multiple rejections of the Metropolis algorithm
at the cost of additional computer storage. We begin with the same data as the
Metropolis algorithm except now we discretize the parameter space into k =1,2, ... ,K z
values. The steps are as follows:
1.
Start with an arbitrary zk on each cell and calculate the initial energy. Set the
initial temperature equal to this energy.
2.
Beginning with the first cell, replace the original zk with each of the k=l,2, ... ,K
values and calculate the energy for each. There are now K energy values
E 1,E2·····Ek; this is equivalent to there being K fields, each differing only in the
value for the first cell.
3.
Calculate the discrete cumulative Gibbs probability for each of the K fields
according to
k
I.exp(-Emfn
m=l
Fk= K
Lexp(
-EmfT)
m=l
4.
Generate a random number r between 0 and 1 from a uniform distribution.
5.
Accept the energy value in the first cell that has the Gibbs probability closest to r.
6.
Test for convergence. That is, test that the energy is below some preset tolerance.
If it is, the procedure is over for this cell; if not, repeat steps 2 to 6 for the next
cell.
7.
If convergence has not been attained when all of the cells have been visited, lower
the temperature and begin again with step 1 at the first cell.
Chap. 12 Modeling Geological Media
326
This procedure has the same number of rejected values (K- 1) each cycle, so, while it also
can be computationally intensive, its operation count is less than for the Metropolis
algorithm. The somewhat artificial discretizing of the z values is less of a problem than
it first appears because the granularity (the increment between successive zi) can be
adjusted.
12-6 CONCLUDING
REMARKS
The technology of statistics, particularly as applied to fluid-flow modeling, has made
remarkable advancements. We know that using statistics to assign the input to a
simulator improves the efficiency of the modeler. We must verify, however, that the
procedures generate realistic heterogeneity and can be made to agree with geology; they
are adjuncts to, not replacements for, geology. The next chapter has further discussion.
These considerations make it likely that the practice of using statistics to assign input
will be standard technology in the near future. But there is one other argument, not yet
realized, that may supersede in importance all of the previous ones.
We continually observe (and occasionally contribute to) the ongoing revolution in
computer technology. What is possible today-a few million cells in a numerical
simulation-was only a dream a few years ago, and with this capability comes a
significant burden in knowledge management. A typical numerical simulation requires at
a minimum three permeabilities, a porosity, a fluid saturation, and a pressure for each
cell. For a one-million-cell simulation, then, six million numbers are required. Simply
assigning these numbers for the simulation is a significant task, and adjusting specific
ones for the purpose (say) of history matching becomes impossible.
On the other hand, each parameter field can be efficiently controlled with an
appropriate semivariogram; thus, instead of six million numbers, we control the
simulation with only 12, if a two-parameter semivariogram is used and the entire field is
one geologic classification. Twelve parameters represent a nice balance between
flexibility and what can be realistically achieved. Thus, it is far more efficient to assign
fields with statistics than to assign the numbers directly. This advantage may be the
most important one for future application and one that will be present even if the current
trend in increasing computer power continues.
13
In this chapter, we develop and apply the concepts introduced in previous chapters for the
modeling of reservoir and aquifer properties. The first part (analysis) draws together data
and statistical concepts into a method for inferring a model for subsequent use in
numerical reservoir simulation. The case studies are arranged in increasing sophistication
of statistical treatment. In the second part (application), we give examples of how
statistical/geological models perform in numerical simulation.
We should remember that the appropriateness of a model depends on the ultimate
application; more sophistication and complexity do not always result in a "better" model.
For example, a tank-type material-balance model of a field can yield a fairly accurate
estimate of reserves and may, in certain circumstances, even provide as much insight as a
more sophisticated model (Dake, 1994, p. 4). Moreover, there are times when a crude
but easy-to-use model will fit better into time and budget constraints. The question in
these cases is not whether a detailed model is better than a simple one, but rather how
much better. However, more detail always yields a better answer, time and budget
notwithstanding. For detailed models, it is how well they capture the essential physics
and geology (the geoengineering) that determines their accuracy.
For most properties, both recognizable (essentially deterministic) geological elements
at some scale(s) and a random element are present. The random element can represent
327
328
Chap. 13 The Use of Statistics in Reservoir Modeling
natural variability, sparse sampling, noisy data, or a combination of all. Stochastic
modeling is a useful method that can be applied to media that are neither entirely
deterministic nor random.
Chapter 1 introduced three types/descriptions of reservoir structures. We repeat these
here with more geologic elaboration.
Strictly deterministic: Correlatable and mappable reservoir units with well understood
internal sedimentary architecture and predictable petrophysical properties.
Mixed deterministic and random (i.e., stochastic): Correlatable and mappable
reservoir units with an internal sedimentary architecture and/or petrophysical
properties defined by vertical and lateral spatial statistics (e.g., facies PDFs, porosity
CDFs, and associated semivariograms).
Purely random: No apparent geological explanation for the distribution of
petrophysical properties in a heterogeneous reservoir.
These categories are listed in order of the amount of information required for modeling.
The deterministic style requires the most information; measurements at all geologically
relevant scales are needed and the geological units must be pieced together. On the other
hand, the purely random style needs only estimates of the mean and variance of the
important flow properties to produce a model. The result of the lack of information in
the random and stochastic cases is that many different but statistically similar realizations
can be produced. The deterministic case produces only one realization.
Reservoirs and aquifers are deterministic only in the most qualitative sense; on the
other hand, strict randomness is extremely rare in nature. However, as the cases below
show, statistics has a role in all three categories. In fact statistics, used in tandem with
geoscientific concepts for reservoir modeling, provides power and authenticity not
achievable by either method alone.
The degrees of determinism and/or randomness, which can be assessed by careful
geological description and sampling, will determine the appropriate modeling strategy.
While a reservoir or aquifer may fall into different categories at different scales, we will
concentrate on the flow unit (Hearn et al., 1984) or interwell scale (over 1 or2 km).
13~1
THE IMPORTANCE OF GEOLOGY
Questions concerning the role of geology often arise during modeling studies. Does the
geology matter and, if so, by how much? While the interaction of heterogeneity and flow
processes is still being researched, there are increasing indications that geological
Statistics for Petroleum Engineers and Geoscientists
329
information may be quite important for accurate flow behavior models. The Page
Sandstone is a case of particular interest and it, along with others, is presented below.
A succession of works has detailed the geology (Chandler et al., 1989), the
permeability distribution (Goggin et al., 1988; Goggin et al. 1992) and fluid flow (Kasap
and Lake, 1990) through the Page Sandstone, a Jurassic eolian sand that outcrops in the
extreme of northern Arizona near Glen Canyon Dam.
The geological section selected for the flow simulation was the northeast wall of the
Page Knob (Goggin, 1988). The data were derived from probe permeameter
measurements taken on various grids and transects surveyed on the outcrop surfaces,
together with profiles along a core. The individual stratification types comprising the
dunes, i.e., grainfall and windripple, have low variabilities (Cv's of 0.21). The interdune
element has a Cv of 0.81. The interdune material is less well-sorted than the dune
elements. Grainfall and windripple elements are normally distributed, the interdune lognormally. The individual stratigraphic elements in this eolian system are well-defined by
their means, Cy's, and PDF's.
The vertical outcrop transects are more variable (Cv = 0.91) than the horizontal
transects (Cv= 0.55), an anisotropy that seems to be typical for most bedded sedimentary
rocks. The global level of heterogeneity for the Page Knob is probably best represented
by the transect along the core, which had a Cv = 0.6. Semivariograms were calculated
for the grids and core profiles. The grids allowed spherical semivariogram ranges to be
determined for various orientations. These ranges indicate the dip of the cross beds ; the
ranges were 17m along the bed and 5 m across the bed (Goggin, 1988). Hole structures
are present in most of the semivariograms, indicating significant permeability cyclicity
that corresponds to dune crossbed set thicknesses.
For our purposes, the most important facet of this work is the modeling of a matcheddensity, adverse-mobility-ratio miscible displacement through a two-dimensional cross
section of the Page Sandstone. Figure 13-1 shows a typical fluid distribution just after
breakthrough of the solvent to the producing well on the right. The dark band is a
mixing zone of solvent concentrations between 30% and 70%. The region to the left of
this band contains solvent in concentrations greater than 70%, to the right in
concentrations less than 30%. The impact of the variability (Cv = 0.6) and continuity can
be seen in the character of the flood front in the large fluid channel occurring within the
upper portion of the panel. There is a smaller second channel that forms midway in the
section. As we shall see, both features are important to the efficiency of the
displacement.
330
Chap. 13 The Use of Statistics in Reservoir Modeling
c
Figure 13-1. Distribution of solvent during a miscible displacement in the detailed
simulation of the Page Sandstone outcrop. From Kasap (1990). The flow
is from left to right.
The simulation in Fig. 13-1 represents one of the most geologically realistic,
deterministic, numerical flow simulations ever run because it attempts to account for
every geologic detail established through prior work. Specifically,
1.
permeabilities were assigned according to the known stratification types at every
gridblock;
2.
the permeability PDF for each stratification type is well-known;
3.
a random component was assigned to the permeability of each gridblock to
account for variance of the local PDF's;
4.
crossbedding was accounted for through the assignment of a full tensorial
permeability in about one-third of the gridblocks;
5.
the specific geometry of the features was accounted for through the use of finite
elements; and
6.
each bounding surface feature (low permeability) was explicitly accounted for
with at least one cell in each surface.
In all, over 12,000 finite-element cells were required to account for all of the geologic
detail in this relatively small cross section. Indeed, one of the purposes of the simulation
was to assess the importance of this detail through successively degraded computations.
See Kasap (1990) for more details.
Statistics for Petroleum Engineers and Geoscientists
331
Another purpose of the deterministic simulation was to provide a standard against
which to measure the success of conditional simulation (CS) as was described in Chap.
12. Figure 13-2A shows two CS-produced realizations of solvent flowing through the
Page cross section. Compared to Fig. 13-1, the permeability distribution was constructed
with a significantly degraded data set that used only data at the two vertical wells
(imagined to be on the edges of the panel) and information about a horizontal
semivariogram. The field on which the simulation of Fig. 13-2A was constructed used a
spherical semivariogram. Figure 13-2B shows the same field constructed with a fractal
semivariogram.
We compare both figures to the distribution in Fig. 13-1. Qualitatively, the fractal
representation (Fig. 13-2B) seems to better represent the actual distribution of solvent; it
captures the predominant channel and seems to have the correct degree of mixing. The
spherical representation (Fig. 13-2A) shows far less channeling and too much mixing
(that is, too much of the panel contains solvent of intermediate concentrations).
However, a quantitative comparison of the two cases shows that this impression is in
error (Fig. 13-3). The distribution that gave the best qualitative agreement (Fig. 13-2B)
gives the poorest quantitative agreement; Such paradoxes should cause us concern when
performing visual comparisons; however, there is clearly something incorrect about these
comparisons. The key to resolving the discrepancy lies in returning to the geology.
Figure 13-2A. Simulated solvent distribution through cross section using a spherical
semivariogram. From Lake and Malik (1993). The mobility ratio is 10.
The scale refers to the fractional (at the injected) solvent concentration.
Flow is from left to right.
·
332
Chap. 13 The Use of Statistics in Reservoir Modeling
0.0
Figure 13-2B.
o.·s
Solvent distribution through cross section using a fractal semivariogram.
From Lake and Malik (1993). The mobility ratio is 10. The scale refers
to the fractional (at the injected) solvent concentration. Flow is from left
to right.
1.0
Spherical semivario!!:r~m.
A.=178m
Reference case
Fractal semivariogram, H:0.29
0.0 0
Figure 13-3.
1
2
Cumulative Pore Volumes Injected
3
4
Sweep efficiency of the base and CS cases. From Lake and Malik (1993).
333
Statistics for Petroleum Engineers and Geoscientists
Figure 13-4 shows the actual distribution of stratification types at the Page Sandstone
panel based on the northeast wall of the Page Knob.
~r
".:t. .·,~-· ;t.-.~, ~·-···-"' ~"'"~'·"~~
·»~,p.r~:£·".s::"f
t..'i.·.(.(:~.:: ..
~~.,::~.'k~·~
.t:~:s~-t~:~.
Figure 13-4.
Spatial distribution of stratification types on the northeast wall of the Page
Sandstone panel. From Kasap and Lake (1993). The thin bounding
surfaces are black lines, the high-permeability grainflow deposits are
shaded, and the intermediate-permeability windripples are light regions.
The thin bounding surfaces are readily apparent (black lines), as are the highpermeability grainflow deposits (shaded) and the intermediate-permeability windripples
(light). This is the panel for which the simulation in Fig. 13-1 was performed. Even
though the entire cross section was from the same eolian environment, the cross section
consists of two sands with differing amounts of lateral continuity: a highly continuous
upper sand and a discontinuous lower sand. Both sands require separate statistical
treatment because they are so disparate that it is unlikely that the behavior of both could
be mimicked with the same population statistics. (Such behavior might be possible with
many realizations generated, but it is unlikely that the mean of this ensemble would
reproduce the deterministic performance.)
When we divide the sand into a continuous upper portion, in which we use the fractal
semivariogram, and a discontinuous lower portion, in which we use the spherical
semivariogram, the results improve (Fig. 13-5). Now both the predominant and the
secondary flow channels are reproduced.
334
Chap. 13 The Use of Statistics in Reservoir Modeling
Figure 13-5. Solvent distribution through cross section using two semivariograms.
From Lake and Malik (1993). Shading same as in Fig. 13-1.
Now both the predominant and the secondary flow channels are reproduced. More
importantly, the results agree quantitatively as well as qualitatively (Fig. 13-6).
The existence of these two populations is unlikely to be detected from limited well
data with statistics only; hence, we conclude that the best prediction still requires a
measure of geological information. The ability to include geology was possible because
of the extreme flexibility of CS. The technological success of CS did not diminish the
importance of geology; rather, each one showed the need for the other.
13-2
ANALYSIS AND INFERENCE
Petrophysical properties (e.g., porosity and permeability) are usually related to the
manner in which the medium was deposited. In fact, it is often observed that textural
characteristics of sediments (e.g., grain size and sorting) play an important role in
determining levels and distributions of petrophysical properties (Panda and Lake, 1995).
The hierarchical nature of sediments is such that reservoirs often comprise systematic
groupings and arrangements of depositional elements. Sedimentology takes advantage of
knowledge of this organization to interpret sediments in terms of depositional
environment. The environments contain a great deal of variability in all aspects, but
335
Statistics for Petroleum Engineers and Geoscientists
"typical" elements exist and the architecture (stacking patterns) of those elements reveals
the processes at work during deposition.
Statistics should be used within the geological framework provided by stratigraphic
concepts (e.g., sequence stratigraphy, genetic units, architectural elements). An
important aspect of the statistical analysis is to detect and exploit the geological elements
from a set of data by using appropriate sampling plans, spatial analysis, and descriptive
statistics.
1.0
0.8
Two semivariograms,
first realization.
Two semivariograms, second realization.
Reference case
0.0 ()
1
2
3
4
Cumulative Pore Volumes Injected
Figure 13-6.
Sweep efficiency of the base and dual population CS cases.
From Lake and Malik (1993).
We now present several case studies that cover the range of reservoir types previously
outlined. The examples show the statistical/geological interplay needed to model a
reservoir.
Rannoch Formation, North Sea
The Rannach formation (Middle Jurassic, North Sea) is a shoreface sandstone reservoir
that occurs as a well-defined flow unit over a wide area. The reservoir unit is generally
continuous over the producing fields, but it is internally heterogeneous. The Rannach
can be distinguished from the overlying Etive formations by careful analysis of porosity-
Chap. 13 The Use of Statistics in Reservoir Modeling
336
permeability data; these show a unit with a left-skewed but dominantly unimodal PDF
(Fig. 3-11). The fine grain size of the Rannoch is largely responsible for the low
permeability compared to the Etive (Fig. 13-7).
GEOLOGICAL
DESCRIPTION
Small scale ~
trough cross
bedding
ETIVE
Wavy bedded,
rippled and
laminated
Low angle cross
laminated
RANNOCH
Low angle cross
laminated
interbedded with ~
ripple laminated ~
-
m~
Medium grained,
bioturbated
\J~v
--
I
I--------·
: BROOM
fmc
grain size
Figure 13-7. Typical Rannoch formation sedimentary profile. f, m, and c
refer to fine, medium and coarse grain sizes, respectively. The
Rannach is generally micaceous (m), and this controls the
small-scale permeability contrasts.
The permeability Cv varies from 0.7 (in Statfjord field) to 1.48 (Thistle field). From
Fig. 6-2, the formation is heterogeneous to very heterogeneous. Carbonate concretions
are present in the Thistle field reservoir, and these can be readily identified as a separate
Statistics for Petroleum Engineers and Geoscientists
337
population from the uncemented reservoir intervals by their low porosity and
permeability (Fig. 10-10). With the concretion data omitted, the permeability Cvreduces
in the Thistle field to 0. 73 to 0.87. The permeability of the Rannach is heterogeneous
because of a subtle coarsening-up profile and small-scale depositional fabric.
There are three possible ways to model the Rannoch permeability patterns. Most
simply, it would be possible.to select the arithmetic average permeability and model the
Rannach as a layer cake with a single effective property. Second, the Rannoch could be
represented with a linear, upward-increasing permeability trend to reflect the gentle
coarsening-upwards grainsize profile. However, both of these deterministic models may
be inadequate if the small-scale heterogeneity affects the flow. The third method is
described further below.
Bedset
10000
30m
0
0
1000
Plug permeability , mD
Cv=0.9
Figure 13-8.
1000
1000
Probe permeability ,mD
Probe permeability ,mD
Heterogeneity at different scales in the Rannach formation. The laminaset packages that occur in the wavy bedded and HCS beds have different
levels of heterogeneity and vertical lengths. For the purposes of
illustration, only certain intervals are shown; for more details and
discussion, the reader is referred to Corbett (1993).
The Rannach is a heterogeneous reservoir (Cv> 0.5), whose heterogeneity varies with
scale and location in the reservoir. For example, at the small scale, the low-mica
Chap. 13 The Use of Statistics in Reservoir Modeling
338
lamination is relatively uniform (Cv = 0.2) in comparison with the wavy bedded material
(Cv = 0.6) over similar 30-cm intervals (Fig. 13-8). At the bed scale, there is significant
heterogeneity in the wavy bedded and hummocky cross-stratified (HCS) intervals. Probe
permeameter data are critical to assess the small-scale variability (Fig. 6-3). The
geological elements at these scales (lamina sets) should behave differently under
waterflood because of the vertical length scales of the geological structure and the effects
of capillary pressure (Corbett and Jensen, 1993b).
Knowledge of the geology aids the interpretation of the statistical measures, and viceversa. Representative vertical semivariograms for the Rannoch show "holes" at 2 em and
1.35 m associated with lamina-scale (high-mica laminated and wavy bedding) and bedscale sedimentary structures (in the HCS), respectively (Fig. 13-9 left and center). The
large-scale semivariogram (Fig. 13-9 right) shows a linear trend over the 35-m Rannoch
interval associated with the coarsening-upward shoreface, with some evidence of ·
additional structure at 10m. These semivariograms are showing signs of deterministic
cyclicity at the lamina, bed, and parasequence scales. The geological analysis is helped
by the spatial measures and, used in tandem, geology and statistics can identify
"significant" (from a petrophysical property point of view) variability.
Increasing vertical scale
0.0
Figure 13-9.
0
0.0
50
100 150 200
Lag, em
0
2000
4000
Spatial patterns of permeability in the Rannoch formation from representative
semivariograms over various length scales. Note the nested hierarchy of
spatial structure and the absence of nugget for the finest-scale data. The
nugget at larger scales represents the poor resolution of the lamina by larger
sampling programs. The semi variance is normalized by the sample variance.
Statistics for Petroleum Engineers and Geoscientists
339
Lochaline Sandstone, lower Cretaceous, West Scotland
The Lochaline mine on the west coast of Scotland provides a rare opportunity for the
collection of three-dimensional permeability data within a reservoir analog (Lewis et al.,
1990). These data have been analyzed and simulated using a variety of statistical
techniques.
The Lochaline Sandstone is a shoreface sandstone as was the Rannach, but the
Lochaline was deposited in a high-energy environment that leads to clean, well-sorted,
large-grain sands. The lateral extent of the properties tends to be more nearly layer-like.
The Rannoch, with its prevalence of HCS bedforms and a mica-rich medium, gives rise
to more permeability and visual contrast at the smaller scale. As we shall see, these
differences appear in the analysis.
f m
Grain size
0
50
(Permeability)l/2, mDl/2
Figure 13-10. Coarsening-up profile in the Lochaline shoreface sandstone.
The coarsening-up profile is a characteristic of prograding
shorefaces (c.f., Rannach profile in Fig. 13-7) albeit much
thinner in this example. The histogram shows a "bell-shaped"
distribution in the square-root domain (i.e., root normal).
The grain-size profile (~ig. 13-10 left) shows a distinct upward-increasing trend in
permeability. The Lochaline data are root-normally (p = 0.5) distributed with a
(heterogeneous) Cv = 0.63 (Fig. 13-10 right). In this case, small-scale laminae and bed
structure were not visible to the eye; the variability at this scale was small (Cv = 0.25)
340
Chap. 13 The Use of Statistics in Reservoir Modeling
and the permeability normally distributed. This is quite different from the Rannoch
shoreface, where the mica contributes to very visible lamination.
At the parasequence scale, the vertical semivariogram shows a gentle linear trend (Fig.
13-11 top). The hole in the horizontal semivariogram at 70 m is probably a result of
difficulties maintaining a constant stratigraphic datum rather than any lateral structure
(Lewis, personal communication, 1994). The nonzero nugget is more likely to be caused
by heterogeneity below the measurement spacing than measurement error, as mm-spaced
data commonly show zero nugget (Fig. 13-9 left). A small nugget was seen in 2-cmspaced data, which suggests that there may be small-scale structure that has gone
undetected. Careful review of the semivariograms is a useful diagnostic procedure.
-·-=
f"-1
2.5xio4-
-55
VERTICAL
f "')
~
~
y
·-....=""'...
0
1
3
2
~
~
1x106-
e
(j,j
00
0.5xi0°
HORIZONTAL
0
100
200
Lag,m
Figure 13-11. Vertical and horizontal semivariograms for probe flow rate
(proportional to permeability) in the Lochaline mine. The
horizontal semivariogram suggests a correlation length of ~bout
50 m with a hole at 70 m. Note the differences in scales
between the two plots.
341
Statistics for Petroleum Engineers and Geoscientists
San Andres Formation Shelf Carbonates,
Permian Basin, Texas
In this example, we consider a carbonate reservoir on which a statistical study has been
carried out (Lucia and Fogg, 1990; Fogg et al., 1991). Flow units extend between wells
but no internal structure can be easily recognized or correlated. The San Andres
formation carbonates are very heterogeneous with a Cy of 2.0 to 3.5 (Kittridge et al.,
1990; Grant et al., 1994). A vertical permeability profile (Fig. 13-12) from a wirelinelog-derived estimator shows two scales of bedding .
.01
1
100
10000
Permeability, mD
Figure 13-12. Permeability profile for the San Andres carbonate generated
from a porosity-based predictor. There are two types (A and
B) of heterogeneity at the bed scale. From Fogg et al. (1991).
A vertical sample semivariogram (Fig. 13-13) shows aspherical behavior out to a
range of about 8 ft. This range appears to be between the scales of the two bedding types
(A and B) shown in Fig. 13-12. The spherical model fit to the sample semivariogram
342
Chap. 13 The Use of Statistics in Reservoir Modeling
overlooks the hole (at around 24 ft) that may be reflecting Type B scale cyclicity. The
structure of Type A beds may not be reflected in the sample semivariogram because of
the effect of the larger "dominating" Type B cycles.
Horizontal semivariograms were derived from permeability predictions generated
from initial well production data (Fig. 13-14). Because of the sparse data, these
semivariograms alone are indistinguishable. But there is a mapped high-permeability
grainstone trend associated with reef development that is consistent with an anisotropic
autocorrelation structure. The geology is critical to interpreting these semivariograms
(refer to Fogg et al., 1991, for further discussion).
8
16
24
32
40
lag,ft
Figure 13-13. Spherical model fitted to a vertical sample semivariogram. The
spherical model ignores the hole at 24 ft that indicates cyclicity.
From Fogg et al. (1991).
13~3
FLOW MODELING RESULTS
As a tool for generating the input to reservoir flow simulations, statistical techniques
offer an unprecedented means of generating insights into reservoir processes and
improving efficiency. We give a few illustrations in this section. In general, we present
results of two types: result~ of fluid-flow simulations that have used statistical modeling
to generate their input permeability or maps of the input itself. All of the cases to be
discussed have used some form of conditional simulation (Chap. 11), and, except where
noted, all have used random field generation in conjunction with geological information.
Statistics for Petroleum Engineers and Geoscientists
343
History Matching
Figure 13-15 shows calculated (from a numerical simulator) and actual results of water
cut from a waterflood in a U.S. oilfield.
2.0
-
<:'1
Q
-
5
Q)
y
-
·-'="' ~
-
~
>
·s
~
00.
0.0
0
Distance, ft
3000
Figure 13-14. Sample semivariograms in two horizontal directions for the San
Andres carbonate. Semivariograms with different ranges can express
the anisotropy suggested by the geological model. From Fogg et al.
(1991).
The calculated results used a hybrid streamline approach in which the results of
several two-dimensional cross sections were combined into a three-dimensional
representation. The permeability distributions in the cross sections were generated by
CS, using, in this case, a fractal representation that was derived from analysis of wireline
logs and permeability-porosity transforms. The calculated and actual results agree well.
The process of bringing the output of a numerical simulator into agreement with actual
performance is known as history matching. History matching is a way of calibrating a
simulator before using it for predictions. In most cases, history matching requires more
than 50% of a user's time, time that could be better spent on analysis and prediction. For
the case illustrated in Fig. 13-15 (and in several other cases in the same reference), the
indicated history match was attained in only one computer run. Historical data are best
used to test and diagnose problems in the model rather than to determine the model.
344
Chap. 13 The Use of Statistics in Reservoir Modeling
Subsequent experience has shown that such agreement is not so easily attained for
individual wells and, even for entire field performance, some adjustment may be needed.
However, the point of Fig. 13-15 is that CS significantly improved the efficiency of the
analyst.
100
Field data
80
+.1
c:
Q)
Calculated
60
~
:.
..s
:::1
40
......i·.
u
.1!)
...
as
s:
20
......
0
0
10
20
30
40
50
Time, Years
Figure 13-lS.Actual and simulated production from a waterflood. From
Emanuel et al. (1989).
Geologic Realism
Figure 13-16A shows a three-dimensional permeability distribution from a carbonate
reservoir in east Texas generated using CS and fractal semivariograms. The dark shading
indicates low-permeability regions and the small circles indicate well locations.
This realization was generated with substantial geological insight, the most important
of which was the identification of six separate facies, indicated in the figure as layers.
Only the frrst two facies (layers 1 and 2) were statistically distinct from the others (Yang,
1990), but the properties within each were nevertheless generated to be consistent with
the respective permeability PDF.
Subsequent analysis revealed the existence of a pinchout (a deterioration in reservoir
quality) to the southeast in this field. The pinchout is represented by the solid lines in
Fig. 13-16B. Such features are not a standard in field~generation procedures and
modeling; however, with some effort they can be incorporated.
Statistics for Petroleum Engineers and Geoscientists
34
3
2
1
0
-1
-2
-3
Figure 13-16A. Three-dimensional permeability realization of an east Texas fiel<
From Ravnaas et al. (1992). (Top row contains layers 1,2, and 3).
3
2
0
-1
-2
-3
Figure 13-16B.
Three-dimensional permeability realizations of an east Texas field wi1
pinchout. From Ravnaas et al., (1992).
346
Chap. 13 The Use of Statistics in Reservoir Modeling
As discussed in Yang (1990), the pinchout is introduced by a two-level CS. The first
level constructs the pinchout along the given line using data along the vertical line from
the rightmost well in the figure. This well, a poor producer, has properties that are
representative of those along the pinchout. After the first level, the second-level CS
proceeds, now using data from the first level as conditioning points (some call these
points "pseudowells"). The result is a gradual but sporadic deterioration of reservoir
quality to the southeast in all layers, as well as a general alteration of permeability in all
layers.
Another way to impart geological realism is through the use of pseudoproperties.
These properties are the usual transport properties-permeability, relative permeability,
and capillary pressure-that have been adjusted so that Darcy's law applies at the scale of
interest. The adjustment takes into account the underlying geology of the field, the flow
properties, and the prevailing boundary conditions.
Corbett and Jensen (1993b) simulated flow through three types of laminae in the
Rannach sets (Fig. 13-7), extracted pseudoproperties from these, and then applied the
pseudoproperties to simulate bedform flow. They found the bed pseudoproperties to be
insensitive to small changes in the bed geometries. The sophistication of their model
could be increased by a series of statistical realizations at each stage that would reflect
the variability and geometry of the beds. This would significantly add to the cost and
complexity of the model exercise, however. The deterministic approach to modeling is
useful for understanding the process and evaluating sensitivities.
To model the upward increase in permeability and the lateral variability in the
Lochaline formation (Fig. 13-10), simple polynomial curves were fitted to the medians
of a series of profiles for each height above a datum. The median was chosen because it
is a robust measure of central tendency. Each median was derived from more than 20
profiles; the residuals provided a PDF from which a random variation was drawn and
added to the deterministic trend in the simulation model. However, the small-scale
structure effects, evaluated by simulations, were found to be insignificant (Pickup,
personal communication, 1993).
The resulting model was used to study a simulated waterflood using a mobility ratio of
10 at a rate of 0.6 m/d. The effect of water preferentially following the high permeability
at the top of the shoreface is clear in Fig. 13-17. In this reservoir analog, the production
mechanism (i.e., waterflood) is most sensitive to the deterministic trend. At the scale of
the system, viscous forces dominate (the water preferentially follows the high
permeability) because there is little evidence of gravity "pulling" the water down into the
bottom of the shoreface. Capillary forces are generally weak in this case because of the
low heterogeneity and the relatively high permeability.
Statistics for Petroleum Engineers and Geoscientists
347
1
4m
..........
~_.;....----150m------..
..,..~
Figure 13-17.
Saturation distribution in a simulated waterflood of the Lochaline
Sandstone. High water saturations are the black region to the left;
the gray region to the right is at initial oil saturation.
In this example, the system was relatively simple but the ability to combine the
deterministic element (increasing upward permeabilities) with a random element
(residuals about the trend) demonstrates the power of stochastic models. The procedure
described in the Lochaline study can be applied to the various scales of interest described
in the Rannach study.
Conditional simulations (conditioned on the well data) of the permeability field in the
San Andres formation were generated to investigate the influence of infill drilling (Fig.
13-18). In this case, the permeability field was modeled as an autocorrelated random
field using single semivariograms in the vertical and horizontal directions. Both
realizations show the degree of heterogeneity and continuity needed for this type of
application. However, there are still deficiencies in the characterization that might prove
impottant.
.1.
The profile in Fig. 13-18 suggests a reservoir dominated by two scales of beds. In
the realizations in Fig. 13-18, it is difficult to see that this structure has been
reproduced.
2.
The autocorrelated random field model does not explicitly represent the baffles
caused by the bed bounding surfaces; these may have a different flow response
from that of the models in Fig. 13-12.
Chap. 13 The Use of Statistics in Reservoir Modeling
348
REALIZATION NO. 192
PERMEABILITY SIMULATION 43-36-A
REALIZATION NO. 90
PERMEABILITY SIMULATION 43-36-A
80 ,-----~==
60
40
20
Distance in Feet
IWM::il 1o.o - 1.o md
ABOVE 1 00.0 md
Ill 100.0-10.0 md
D
Figure 13-18.
1.0-0.1 md
BELOW 0.1 md
Two realizations of the spatial distribution of the permeability field for
the San Andres generated by theoretical semivariograms fit to the
sample semivariograms shown in Figs. 13-9 and 13~10. The
permeability fields are the same in the two realizations for the extreme
left and right ends of the model where the conditioning wells are
located. From Fogg et al. (1991).
The panels in Fig. 13-18 also illustrate a generic problem with pixel-based stochastic
modeling: there is no good a priori way to impose reservoir architecture. If the
Statistics for Petroleum Engineers and Geoscientists
349
geometry of the beds were determined to be important, other modeling techniques (e.g.,
object-based) might have been useful.
The San Andres study illustrates how semivariograms can be used to generate
autocorrelated random fields that are equiprobable and conditioned on "hard" well data.
These realizations can form the basis of further engineering studies; however, it is
unlikely that all 200 realizations will be subjected to full flow simulation. Usually only
the extreme cases or representative cases, often selected by a geologist's visual
inspection, will be used.
The main point from these exercises is that geostatistics and CS procedures are
extremely flexible; if geology is known, it can be honored with suitable manipulation
both as a preprocessing step (previous section) or in postprocessing.
Hybrid Modeling
If the reservoir lithology can be discriminated into two populations, for example by a
bimodal PDF of a particular wireline-log measurement, the autocorrelation can be
determined using the resulting discrete lithological indicator.
Rossini et al. (1994) have taken this approach with a sandy dolomitic reservoir where
a wireline-log value has been taken to distinguish sandy facies ("good" reservoir) from
dolomitic facies ("poor" reservoir). Sequential indicator simulation (SIS) using faciesindicator semivariograms is used to generate the model facies fields. Porosities within
each facie are then assigned as correlated random fields from the appropriate faciesspecific semivariograms using sequential Gaussian simulation (SGS). The permeability
is determined from the porosity data using a regression relationship and a Monte Carlo
procedure. The resulting fields capture both the variability and architecture of the
geological structure (i.e., facies architecture), without recourse to a large data base of
channel attributes (needed for object modeling). However, indicator semivariograms are
needed, and these often must be imposed from an understanding of the geological
structure in the absence of real data.
In this study by Rossini et al. (1994), the quality of the statistical models was assessed
by a grading system based on the maximum number of water-cut matches to production
data. The high-graded realizations were then used for full-field simulations to develop a
management strategy. This interesting case study combines many of the statistical
treatments covered in this text-PDF's, transformations, regression, spatial correlation,
and statistical modeling-and provides a useful introduction for all students of reservoir
geo-modeling.
350
Chap. 13 The Use of Statistics in Reservoir Modeling
Fluvial reservoirs, in which the flow units can comprise reservoir units (channel
sandstones, crevasse splay sandstones) and nonreservoir units (lagoonal muds, coals) are
often modeled with object-based techniques (using the known or inferred geometry of
the sands and/or shales). The use of object-based models for the simulation of low
net:gross reservoirs (e.g., Ness formation, Upper Brent, North Sea) have been discussed
in Williams et al. (1993). These systems have also been modeled using a marked-point
process (Tyler et al., 1994), where the "marked points" represent the principal axis of a
channel belt to which the modeled channel attributes are attached. In both techniques,
discrete architectural elements are distributed in a background matrix.
The realism of the models depends on how well the input parameters (e.g., total
reservoir net:gross, channel belt azimuth, and channel belt width:thickness ratios) can be
defined. Net:gross can be obtained from well information; width-thickness data rely on
appropriate outcrop analog data~ Other data sources include "hard" data (e.g., well logs
and well test) and "soft" data (e.g., regional channel orientations or maps of seismic
attributes) to further condition the models.
The advantage of the statistical techniques is that numerous "equiprobable"
realizations can be generated. These realizations are data-hungry and the need for
screening prior to use in an engineering application can limit the usefulness of all the
realizations. Often the connectivity alone is of major concern and this can be determined
relatively simply (Williams et al., 1993). The balance between models and realism has to
be carefully weighed so that the uncertainty in the engineering decision-making process
is adequately quantified within the time frame and budget available.
Fluid-Flow Insights
Numerical simulation has been in common practice for several years, and we have come
to believe in the correctness of its results if we are confident in its input. This procedure
requires careful geological and statistical assessment. Now, we take a different approach
and, using hypothetical models, examine the sensitivities of recovery processes to
changes in the statistical modeling. Statistical modeling through CS offers a way to
control both the heterogeneity and continuity of an input flow field with sufficient
generality to gain novel insights into reservoir processes.
We have known for several decades that miscible displacements in which the solvent
is less mobile than the phase being displaced have a tendency to form an erratic
displacement front, bypass, and, as a consequence, inefficiently recover the resident fluid.
The ratio of mobilities of such a displacement (the mobility ratio) being greater than one,
the displacement is termed adverse. This phenomenon, among the most interesting in all
of fluid dynamics, has been modeled in the laboratory by numerical simulation and
through analytic methods. Nearly all of this work has been in homogeneous media. The
Statistics for Petroleum Engineers and Geoscientists
351
question then arises: what happens to adverse miscible displacements in the presence of
realistic heterogeneity?
This was the issue in the work of Waggoner et al. (1992) extended by Sorbie et al.
(1994), who studied adverse miscible displacements through two-dimensional cross
sections of specific heterogeneity and continuity. This work used the Dykstra-Parsons
coefficient VDP (Chap. 6) to summarize heterogeneity and the spherical semivariogram
range (Chap. 11) in the main direction of flow to summarize continuity. The
displacements fell into three categories.
1.
Fingering: displacements in which bypassing occurred, caused by the adverse
mobility ratio.
2.
Channeling: displacements in which bypassing occurred, caused by the
permeability distribution.
3.
Dispersive: displacements in which no bypassing occurred.
Both fingering and channeling displacements manifest inefficient recovery and early
breakthrough of the solvent. The dispersive displacements showed efficient recovery and
late breakthrough. The principal distinction between fingering and channeling is that a
fingering displacement becomes dispersive when the mobility ratio becomes less than
one; a channeling displacement continues bypassing.
These classifications are of little use without some means of saying when a given type
will occur. Figure 13-19 shows this on a flow-regime diagram.
The figure shows some interesting features. For example, it appears possible to
completely defeat fingering with uncorrelated heterogeneity as, for example, might occur
within a carbonate bed (Grant et al., 1994). However, the overwhelming impression is of
the ubiquitousness of channeling. In fact, for typical values of VDP. fingering does not
occur and there are dispersive displacements only in essentially random fields. It appears
that fingering occurs only in regimes where VDP is less than 0.2. Such low values are
normally possible only in laboratory displacements. The transition to channeling flow
normally occurs at a heterogeneity index IH of around 0.2.
The above two observations render a remarkable change in the point of view of
miscible displacements: rather than being dominated by viscous fingering, as had been
thought, they appear to be channeling. Indeed, channeling appears to be the principal
displacement mode for most field displacements. These insights have operational
significance. For example, it seems obvious that decreasing well spacing (increasing the
dimensionless range) will always lead to more channeling and, consequently, less sweep
Chap. 13 The Use of Statistics in Reservoir Modeling
352
and ultimate recovery. Such a loss might be compensated by an increased rate; however,
smaller well spacings will always lead to more continuity between wells.
2.0
M:10
~
•
•
=1.5
~
~
·-cJ."G-... 1.0
Channelling
~
•
•
~
-~ 0.5
e
Q 0.0
0.0
Figure 13-19.
0.2
0.6
0.4
Dykstra-Parsons Coefficient
0.8
1.0
The various types of flow regimes for a matched-density miscible
displacement. From Waggoner et al. (1992). Filled boxes indicate
computer runs.
The Waggoner et al. (1992) conclusions have been extended to unmatched density
displacements by Chang et al. (1994), to more general cases by Sorbie et al. (1994), and
to immiscible displacements by Li and Lake (1994). In the context of this discussion, we
emphasize that such insight can only be possible with a suitably general representation of
heterogeneity afforded by CS.
13~4
MODEL BUILDING
The few case studies presented above illustrate several aspects of reservoir modeling
using statistics. We summarize these examples, in this section, to sketch a method for the
selection of an appropriate approach. Remember, the goal is to generate a three-
Statistics for Petroleum Engineers and Geoscientists
353
dimensional array of petrophysical properties (mainly permeability) that can be used in
numerical simulation.
Geologic Inference
Geology provides several insights that are useful for statistical model-building:
categorization of petrophysical dependencies, identification of large-scale trends,
interpretation of statistical measures, and quality-control on generated models. The first
two serve to bring the statistical analysis closer to satisfying the underlying assumptions.
For example, identification of categories and trends, and their subsequent removal, will
bring data sets closer to being Gaussian and/or to being stationary. The last two are to
detect incorrect inferences arising from limited and/or biased sampling. Consequently,
we shall see aspects of these in the following procedures.
The first two steps are common for all procedures.
1.
Divide the reservoir into flow units. f}.Q~ J.mi~.are packages of reservoir rocks
withsimilar petrophysical characteristics-not necessarily uniform or of a single
rock type-¢..ilUlJ?Pea~ in all or most of the wells. This classification will serve to
develop a stratigraphiC framework ("stratigraphic coordinates") for the reservoir
model at the interwell scale.
2.
Review the sample petrophysical distributions with respect to geological and
statistical representativity.
What follows next depends on the properties within the flow units, the process to
be modeled, and the amount of data available. We presume some degree of
heterogeneity within the flow unit.
3.
If there are no data present apart from well data and there is no geologic
interpretation, the best procedure is to simply use a conditional simulation
technique applied directly to the well data (Fig. 13-20). The advantage of this
approach is that it requires minimal data: semivariogram models for the three
orthogonal directions. (Remember, a gridblock representing a single value of a
parameter adds autocorrelation to the field to be simulated.)
Chap. 13 The Use of Statistics in Reservoir Modeling
354
y
~
~~
f
Column k
h_z---lllilio>.,._
~ ~ h_x_ _....,....,.
km
Figure 13-20.
hy
Generation of a three-dimensional autocorrelated random field using
orthogonal semivariograms. hx, hy and hz are the lags in the coordinate
directions. dm is decameters, km kilometers.
While parameters for vertical semivariograms are usually available from the well
data, the lateral parameters must be estimated from the normally sparse well data
and/or reservoir analogs. It is important that uncertainties in these parameters be
acknowledged in the modeling schemes. This approach is the most stochastic of
those considered here and will lead to the most uncertainty in the predictions.
This approach was used in the Page (Fig. 13-3) and San Andres (Fig, 13-18)
examples.
4.
If, from geological analysis,
a representative geological structure can be
recognized within the flow unit and the flow unit itself is laterally extensive, a
deterministic model can be built. The scales of heterogeneity can be handled by
geopseudo upscaling (Fig. 13-21). The role of statistics in this case is to assess
how representative is the structure. Structure can be identified by nuggets, holes,
and/or trends in the semivariograms. Lateral dimensions rely on the quality of the
geological interpretation in this model. The Rannoch example followed these
steps.
5.
If the property distribution is multimodal, the population in the flow unit can be
split into components and indicator conditional simulation can be used to generate
the fields. This is useful in fields where the variation between the elements is not
clearly defined and distinct objects cannot be measured. This approach
(Fig. 13-22) yields fields with jigsaw patterns. This was the approach taken by
Rossini et al. (1994). See also Jordan et al. (1995).
355
Statistics for Petroleum Engineers and Geoscientists
ylc
~
dm
~c
km
~lc
Column
hz
flilll"'
hx
rh
km
Figure 13-21.
f
....
Y
Schematic reservoir model at two hierarchical scales capturing welldetermined geological scales at each step. A small-scale (em to m)
semivariogram guides selection of representative structure.
'Y
~
Core
~
k"
~ .,... Geopseudos ' cmtom
~
Geology
Deterministic
Model
Figure 13-22.
6.
Schematic of a sequential indicator simulation for a binary system.
Indicator semivariograms are determined from a binary coding of the
formation. hx, hy and hz are the lags in the coordinate directions. dm
is decameters, km kilometers.
If the flow unit contains distinctly separate rock types and a PDF can be
determined for the dimensions of the objects, the latter can be distributed in a
matrix until some conditioning parameter is satisfied (e.g., the ratio of sand to
total thickness). This type of object-based modelling lends itself to the modeling
356
Chap. 13 The Use of Statistics in Reservoir Modeling
of labyrinth fluvial reservoirs (Fig. 13-23). More sophisticated rules to follow
deterministic stratigraphic trends (e.g., stacking patterns) and interaction between
objects (e.g., erosion or aversion) are available or being developed. A similar
model for stochastic shales would place objects representing the shales in a
sandstone matrix following the same method.
Figure 13-23.
Schematic of object model of a fluvial reservoir. Dimensions of
objects are sampled and their locations set by sampling CDF's.
These models are the most realistic of all for handling low net-to-gross fluvial
reservoirs; however, they require a good association between petrophysical
properties and the CDF's for the geometries. Tyler et al. (1995) give a good
example of this application.
If the reservoir contains more than one flow unit, then the procedure must be repeated
for the next zone. In the Page example (Fig. 13-6), the upper layer needed a different
correlation model (generated by a fractal semivariogram) from the lower layer (spherical
semivariogram).
The modelling of petroleum reservoirs requires an understanding of the heterogeneity,
autocorrelation structure (nested or hierarchical) derived from the geological architecture,
and the flow process to select the most appropriate technique. As reservoirs have various
correlation structures at various scales, flexibility in modeling approach must be
maintained. The "right" combination of techniques for a Jurassic reservoir in Indonesia
may be entirely inappropriate for a Jurassic North Sea reservoir. The methods discussed
here try to emphasize the need for a flexible toolbox.
Statistics for Petroleum Engineers and Geoscientists
357
Other Conditioning
As has been the thrust of this entire book, the above section has concentrated on geologic
and well data, the so-called static data. Other data sources abound in engineering
practice and should, as time and expense permit, be part of the reservoir model also. The
most common of these are the following.
1.
Pressure-transient data: In these types of tests, one or more wells are allowed to
flow and the pressure at the bottom of the pumped wells observed as a function of
time. In some cases, the pressure is recorded at nonflowing observation wells.
Pressure-transient data have the decided advantage that they capture exactly how
a region of a reservoir is performing. This is because interpreted well-test
properties represent regions in space rather than points. Resolving the disparity in
scales between the regional and point measurements and choosing the appropriate
interpretation model remain significant challenges in using pressure-transient
data. (See Fig. 12-2 for the results of attempts to incorporate pressure-transient
data into Kriging.) However, this type of data is quite common and may be
inexpensive to acquire, making it an important tool in the tool box.
2.
Seismic data: The ultimate reservoir characterization tool would be a device that
could image all the relevant petrophysical properties over all the relevant scales
and over a volume larger than interwell spacing. Such a tool eludes us at the
moment, especially if we consider cost and time. However, seismic data come
the closest to the goal in current technology.
There are two basic types of seismic information: three-dimensional, depthcorrected traces "slabbed" from a full cubic volume of a reservoir, and twodimensional maps of various seismic attributes. Seismic data are generally
considered as soft constraints on model building because of the as-yet limited
vertical resolution. However, because of the high sampling rate, seismic data can
provide excellent lateral constraints on properties away from wells. Seismic data
integration into statistical models, mainly through co-Kriging and simulated
annealing, are becoming common in large projects (MacLeod et al., 1996)
3.
Production data: Like pressure-transient data, production data (rates and
amounts of produced fluids versus time) reflect directly on how a reservoir is
performing. Consequently, such data form the most powerful conditioning data
available. Like the seismic integration, incorporating production data is a subject
of active research (see Datta-Gupta et al., 1995, for use of tracer data) because it
is currently very expensive. The expense derives from the need to run a flow
simulation for each perturbation of the stochastic field. Furthermore, production
data will have most impact on a description only when there is a fairly large
quantity of it, and, of course, we become less and less interested in reservoir
358
Chap. 13 The Use of Statistics in Reservoir Modeling
description as a field ages. Nevertheless, simulated annealing offers a significant
promise in bringing this technology to fruition.
13-5
CONCLUDING REMARKS
We have seen the basics of statistics, how geology and statistics (and other technologies)
might be used to complement each other in analysis, and how their combinations produce
the best predictions. As we stated in the introduction, what constitutes the "best" model
depends on the data available, the reservoir type, the processes envisioned, and of course
budget and time. Despite ongoing progress, one thing seems clear: statistical methods
(to be sure, along with improved computing power) have vastly improved our ability to
predict reservoir performance. We hope that the technologists from the various
disciplines will be helped by this volume as the science of generating realistic
"numerical" rocks will remain a common ground for geoscientists and statistical research
in years to come.
M
LATU
Normal
[=]means "has units of"; Lis length, t time, m mass, and F force
A,B,C
events
a
arbitrary constant; constant in Archie's law
a,b,c
parameters of a uniform or triangular distribution
b
constant weight applied to element i in a summation; sensitivity
coefficient
bias of an estimator[=] parameter units
c
arbitrary constant
df
degrees of freedom
covariance of two random variables X and Y [ =] units of X times units
of Y
Cov(X,Y)
Cv
ci
coefficient of variation of a random variable
Dp
particle diameter [=] L
Dpi
grain diameter of fragment i [=] L
d
total sampling domain length [=] L
dmax
maximum vertical separation between two sample CDF's
do
sample spacing required for sampling density ! 0 [=] L
cumulative storage capacity
359
360
E(X)
erf(x)
Nomenclature
expected value of a random variable X [==]X
error function of x
c
exp (x)
exponential function; same as eX
F
F statistic or ratio of variances
F(x;(j,J.l)
theoretical normal or Gaussian cumulative distribution function of
random variable X with mean J.l and standard deviation (j
f(x)
probability distribution function of a continuous random variable X
[=]X-1
f(x,y)
joint probability distribution function for two continuous random
variables X andY [=]X -lor y-1
fw
fractional flow of water
fnx
conditional probability distribution function of random variable Y given
a fixed value of the random variable X = x0
f(x; (j,J.l)
theoretical normal or Gaussian probability distribution function with
mean J.l and standard deviation (j [=] x-1
Fx(x) or F(x)
cumulative distribution function (CDF) of a random variable X for a
bounding value x; flow capacity
F(z)
standarized normal or Gaussian cumulative distribution function with
zero mean and unit standard deviation
F,Fa
untruncated and (apparently) truncated or original CDF, respectively
fA.JB
fraction of data points taken from populations A and B, respectively
!b./t
!i
fraction of bottom~ or top-truncated data points, respectively
proportion of smaller grain fragment i derived from larger grain, fraction
or percent; fractional flow of phase i
gravitational acceleration vector[=] L/t2; (scalar symbol g)
thickness of sublayer i in a reservoir zone [=] L
Hurst or intermittency coefficient
alternative hypothesis
Koval heterogeneity factor
null hypothesis
separation or lag distance [=] L
terminal value of index i
Statistics for Petroleum Engineers and Geoscientists
361
total number of ways (combinations) of getting at levels successes in I
trials for a series of binomial experiments
lH
Gelhar-Axness coefficient or heterogeneity index
J
K
terminal vaue of index}
terminal value of index k
k
permeability tensor[=] L2; (scalar symbol k)
kxx• kxy. kxz,
kyx. kyy• kyz,
kzx, kzy, kzz
elements in full permeability tensor [=] L 2
kx, ky, kz
permeability in the x, y and z coordinate directions [=] L 2
kv,kh
vertical and horizontal permeabilities [==] L2
krj
relative permeability of phase j, fraction or percent
L
length[=] L
Lc
ln(x)
Lorenz coefficient
log(x)
base 10 logarithm of x
M
total number of samples taken from a continuous distribution
MSe
mean square error or standard error of regression
N(Ji,cP)
p
Gaussian normal probability distribution function with mean 11 and
variance cfl
probability proportion of the occurence of successful outcomes in a
series of binomial experiments; tolerance about the sample mean,
percent; pressure in Darcy's law[=] F/L2
p
power coefficient in p-normal distribution
Prob(AIB)
probability of the occurence of event A conditioned to the prior
occurence of eventB, fraction bounded by [0,1]
natural (Naperian) logarithm of x
probability of the occurence of event E, fraction bounded by [0,1)
coefficient of determination for two joint normally distributed variables
all real numbers on a line [=] mapped variable
resistivity of formation component i [=) ohm-m
r
multiple factor, cons.tant
r(x,y)
correlation coefficient of two variables x andy
362
s
Nomenclature
number of successful outcomes in a series of binomial experiments
saturation of phase j, fraction or percent
sum of squares defined as the sum of the product of random variable X
deviations from llX times random variable Y deviations from f.LY [ =];
units of X times units Y
sum of squared residuals in response data about the regression model
line [=] squared units of response variable
SSe
residual sum of squares
estimated variablity of sample regression parameter ~0 (intercept)
1\
Sy
estimated total variability of all data in several data sets [=] squared
units of measured property
estimated variability of averages from several data sets [=] squared units
of measured property
1\
sv
estimated standard error of the Dykstra-Parsons coefficient.
s~
sample standard deviation of an estimated parameter
8 [=] parameter
units
sample variance of a variable X [=] squared units of X
Tx
sum of ranks of all Xi samples used in a nonparametric Mann-WhitneyWilcoxon rank sum test
estimated value for student's t distribution; time
t value for regression parameter ~0 (intercept)
t value of student's distribution given a confidence level a and df
degrees of freedom
t( a,df) or t 0
critical value from student's t distribution for level of confidence
df degrees of freedom
u
u
volumetric flux or velocity [=] L/t
Var (X)
a and
volumetric flux vector [=] L/t
variance of random variable X, or the second centered moment of X [=]
x2
363
Statistics for Petroleum Engineers and Geoscientists
Dykstra-Parsons coefficient
estimation function W or method used to estimate one or more
~
parameter(s)
X(m)
8
from a set of data {Xt,X2; ... ,X)}
result of random variable X as a mapping of an outcome
sample space Q onto the real line (R ') [=] random variable
m from the
sample value i from random variable X
Xi
predicted property value at ith recursive iteration in May's equation [=]
predicted property
X,Y
vectors of measured data
X,Y,Z
stochastic variables
x,y,z
deterministic analogs of X, Y, and Z; coordinate directions
Subscripts
0
null
0.16, 0.25, 0.50,
and 0.75
16th , 25th (1st quartile), median, 75th (3rd quartile) percentile of a
random variable
A
alternative; arithmetic mean or average
a
apparent
A,B
populations A or B
b
bottom-truncated
D
dimensionless
G
geometric mean or average
H
harmonic mean or average
I
indicator
i,j, k
indices of a variable with terminal values I, J, K
0
base value of a variable
p
power mean
smui
sandstone
Nomenclature
364
silt
siltstone
T
true mean, or total
true or formation value; top-truncated
w
water phase
Superscripts
-1
inverse function; vector, matrix inverse
c
complementary or complement; critical or cutoff value
m
cementation exponent in Archie's law
n
saturation exponent in Archie's law
0
base value of a variable
*
standard normal variable
p
power exponent
w
estimator W
(1),(2), .. ,(m)
predictor or explanatory variables
T
vector or matrix transpose
Overstrikes
A
estimate of a variable or function
average (usually arithmetic) of a variable
Greek Symbols
p
r
r1. r2
confidence level, fraction or percent
intercept parameter for linear regression model [=] response variable
slope parameter for linear regression model [=] mtio of response variable
units to predictor variable units
vector of slopes for multilinear regression model
semivariance
coefficients of skewness and kurtosis
Statistics for Petroleum Engineers and Geoscientists
365
gradient operator
random error, frequently assumed to beN(O,l) distributed
general parameter
weight applied to ith data value, typically the Kriging estimator or
variance formula
exponential correlogram autocorrelation length (theoretical mean);
nonlinearity coefficient in May's equation
mobility of phase[=] FL/t; (scalar symbol A)
integral scale
spherical semivariogram range
population mean (noncentered moment of order 1)[=] random variable;
viscosity in Darcy's law[=] F/Lt
noncentered moment of order r [=] random variable raised to the rth
power
rth centered moment of a random variable [=) random variable raised to
the rth power
Pythagorean constant= 3.141592.
p
phase density [=] mfr}
Pg
grain density [=] m/1}
p(X,Y)
correlation coefficient of two random variables X and Y
population standard deviation of a random variable [=] random variable
population variance or second centered moment of a random variable [=]
squared units of random variable
variance of random error in response about the regression model line [=]
squared units of response variable
an experiment, the operation of establishing conditions that may
produce one of several possible outcomes or results [=] outcome units
an outcome or element of the sample space n [=].Q
null set, an event E possessing no outcomes or results
porosity of a given rock volume [=] fraction or percent
core-plug, wire-line porosities[=] fraction or percent
probability density function for chi-squared distribution
366
Nomenclature
sample space, a collection of all possible outcomes of an experiment .5,
[=] element units
Math Symbols
E{a,b}
element of a real-valued set bounded by a and b, inclusively
3
complementary event, the subset of the sample space
contain the elements of event E [=]event element units
n
u
intersection
Q
that does not
union
given that or conditioned upon (e.g. YIX=x 0 means Y given that X=
Xo)
Acronyms
BLUE
best linear unbiased estimator
CDF
cumulative distibution function, see F(x)
C.I.
confidence interval
CLT
central limit theorem
gamma ray wireline log response [=] API-GR units
GR
HCS
IQR
hummocky cross-stratification
JND
joint normal distribution of two correlated random variables
MVUE
minimum variance unbiased estimator
OK
ordinary Kriging
interquartile range taken as the difference (Xo.75- Xo.25)
OLS
ordinary least squares
PDF
probability density function; see f(x)
RMA
reduced major axis
RV
SA
random variable
simulated annealing
scs
swaly cross-stratification
SD
SE
SK
sample standard deviation of a random variable [=] random variable
STOOIP
stock-tank original oil in place[=] standard L3
WLS
weighted least squares
standard error
simple Kriging
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375
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Index
additivity
coefficient of variation 143-144
confidence intervals 123
Dykstra-Parsons coefficient 178-180
hypothesis testing 169
1ocall23
analysis of variance (ANOV A)
for comparing sample means 179-180
significance tests for regression using
228-229
Archie's law 115
autocorrelation
definition 263
measures 267-271
models 285, 287
sampling 290-291
average
arithmetic 68, 107, 129-132
geometric 136, 138-139
harmonic 136, 138
power 135-136, 138
Bayes' theorem 32-36
bias
correction using jackknife 125
definition 109-111
geometric average 136
harmonic average 136
uniform PDF endpoints 113-114
binomial distribution 76-78, 81
carbonate 8
modeling 343-344, 346-348, 349-351
variability 146-147
Carman-Kozeny 120-122
censored data 93, 142
central limit theorem 81-82
channeling 162, 13-24
chi-squared (X2) test
for joint normality 196
clastic 8
laminated 255-256, 281-283
massive 184, 256
permeability anisotropy 284
semivariogram behavior281-283, 290
variability 146-147
coefficient of determination 200, 229-231
coefficient of sample correlation
definition 193
significance testing 200
coefficient of skewness 75-76
coefficient of variation
95% limits for 145
and depositional environment 146-147
and geologic scales 146-147
and sampling density 150-151
and sensitivities 149
definition 75, 145-146
for log-normal distributions 87
of geologic elements 147-148
ofmixtures 148-149
coefficient of kurtosis 76
concentration distribution 83
conditional simulation (see simulation)
383
384
confidence
level168-171, 218-230
regions 220-223
test(s) (see hypothesis test)
confidence intervals
applications 123-126
coefficient of variation 145
definition 108
Dykstra-Parsons coefficient 154
interquartile range 135
limitations 228
linear model217
Lorenz coefficient 159-160
new observations about a regression line
225-226
sample mean 132-134
sample median 126-128, 134-135
sample variance 131
using the bootstrap 125-126
using the jackknife 125-128
with probability assignment 123
consistency 115
correlation(s) (see also autocorrelation and
semivariogram)
coefficient 193-195
comparing porosity-permeability
200-205
improving 254-256
oftwo random variables 189,263-264
covariance
matrix 258
of independent random variables 193
of two random variables 192-193
cumulative distribution function (CDF)
class size 42
complement of 41
conditional45
continuous 43, 44
definition 41-44
discrete 42,47
empirical or sample 46-48
log transformed 47
Index
modeling, estimation and diagnosis of
52
properties of 41
transforming 58-61
Darcy's law 14
degrees of freedom
for F test 177, 181, 223,229-230
for no-intercept regression 233
for tt est 172, 175, 218, 225
for two-parameter regression 216
density 237
depositional environments 38
aeolian (or eolian) 80, 328-335
as random variables 38
fluvial 138, 348-349
shoreface 55,281-283, 336-340
submarine fan 138
variability in 146-147, 336-340
deterministic
definition 2
prediction 2
detransformation 242-245
dispersion coefficient 164
and Gelhar-Axness coefficient 165
and Koval factor 165
definition 164
Dykstra-Parson's coefficient 153-157
"best-fit" estimate of 153-154, 157
comparisons 178-180
estimate bias 155
estimate variability 155
sample standard deviation estimate of
155
efficiency
definition 111
for estimating uniform PDF endpoints
114
for median 135
error(s)
additive 79
in explanatory variable 234
input 12
mean square 216-218
Statistics for Petroleum Engineers and Geoscientists
measurement 210
model210
multiplicative 85
numerical 12
random 209-?10
scale-up 12
standard 108
type 1 and type 2 171
error function 81
estimation
definition 14, 108
physical interpretation in 139-140
estimator(s)
and physical interpretation 139-140
biased 109-110
BLUE 218, 295
coefficient of variation 144-145
consistency 115
Dykstra-Parsons coefficient 153-157
efficiency 111
for non-arithmetic means 135-136
Lorenz coefficient 159-160
MVUE 112
of sample mean 129
properties of 109
robustness of 109, 115-116
sensitivity to erroneous data 116-118
trimmed 141-142
weighted 140-142
Winsorized 136,141-142
event(s) (statistical)
combination of 20, 21
complementary 2,4
definition 19
elementary 20
exhaustive sequence of 22, 31
impossible 27
independent 28
intersection of21, 27
mutually exclusive 22, 27, 31
occurence of 20
probability of 23, 25
385
relative frequency of occurence of
23
union of 20-21
expected value(s) or expectation (see also
mean)
of a continuous random variable
67
of a descrete random variable 67
of the product 70
of the sum 69
properties of 68
experiment(s) (statistical)
definition 19
occurence of 20
outcomes of 37
explanatory
power 213, 229
variable 209
exploration risks 28
exponential (Boltzmann) distribution
89,90
F-test
andANOVA 180
for comparing Dykstra-Parsons
coefficients 178-180
for comparing sample means 180-182
for comparing sample variances 176-178
for regression 230-231
limitations 180
facies
discrimination 21
independence of succession 27
modeling 316-317
factor analysis 262
i frequency
and probability 23
definition 20
of occurence 20, 37,41
polygon 42, 43
relative 23, 24
Gelhar-Axness coefficient 161-162
geostatistics 1, 293
Gini's coefficient 159
386
grain
density 24, 25, 252-254 ·
distribution 82-83
size 83, 339
heterogeneity
definition 143-144
dynamic measures of 162-165
static measures of 144-162
histogram
definition 50
detecting modes 51
history matching 52
hypothesis test(s) (confidence test)
additivity requirements of 169-170
alternative (examples) 171, 172, 175
and type 1 or type 2 errors 170-171
definition 168-170
limitations 168-170, 182
nonparametric 182-187
null (examples) 171, 172, 175
parametric 171-182
intercept (see also least-squares, regression)
confidence interval for 218-219
estimates of211-213
independent t-test of 219-221
joint F-test of 223-224
properties of estimates 215-217
interquartile range
definition 46
properties of 135
integral scale 288
joint (bivariate) distribution(s)
definition 189-192
normal (JND) 195-199
Kolmorogov-Smirnov test 185-188
5% two-sided critical value for 186
wireline and core plug comparison
186-188
Koval factor 162-164
and breakthrough time 162
and Dykstra-Parson's coefficient 163
and Gelhar-Axness coefficient 164
and sweepout time 162
Index
Kriging
co-Kriging 311
definition 294-295
disjunctive 311
indicator 305-310
nonlinear constraints 303-304
ordinary 299-303
simple 295-299
universal 311
least-square(s) (see also regression)
intercept estimate from 212
method of 211-213
normal equations 213
ordinary 215
residuals 213, 219, 221, 226-228
slope estimate from 212
weighted 215, 247-248
linear model (see also regression, least-
squares)
confidence interval for mean response of
225-226
of regression 209-210
validity 234
log-normal distribution 83
Lorenz coefficient 158-161
advantages of 159-160
bias and precision of 160-161
definition 158-159
related to Vnp, Cv, and cr 159
Mann-Whitney-Wilcoxon Rank Sum test
176, 183-185
one- or two-sided 183
material balance 249-251
May's equation 5
mean (see also expected value, average)
binomial distribution 77
binomial mixture of two random
variables 78
centered moments 70
coefficient of variation 74
confidence interval for sample 132-134
exponential distribution 89
flow associations 136-137
Statistics for Petroleum Engineers and Geoscientists
geometric 135, 138
Guassian distribution 80
harmonic 136, 138
power 135-136, 138
properties of sample 129, 132-134
triangular distribution 89
uniform distribution 88
variance formula with known 71
variance of sample 73
Winsorized 141-142
measurements and geology 6, 59,254-256
median
definition 46
jackknifing the 127-128
properties of the sample 134-135
miscible displacement 349-350
models
additivity 236-237
appropriateness 327
building 351-354
hybrid 348-349
importance of geology 328-335
inadequacies 210,238
regression 208-210
semivariogram 287-288
validity and geology 235
moments
centered 70
combination of 74
noncentered 67
other 74
Monte Carlo
reserves 53
stochastic shales 43
N-zero method 150-153
normal (Gaussian) distribution 80
and standardized normal variate 81
and the central limit theorem 78- 80
joint or bivariate 195-198
null set
and mutually exclusive events 23, 25
definition 20
oil
387
pressure and recovery 258-261
recovery, incremental258-261
reserves 57- 58
slug size and recovery 258-261
ordinary least squares (OLS) 215
parsimony 207
partitioning data sets 95-100
for intersecting parents 99, 100
for nonintersecting parents 96, 97
into two populations 97, 98
type curves for 101
permeability
and flow properties 136-138, 170
definition 340
directionall84
histograms and geology 52-54
hypothesis tests 169
log-normal PDF 82-83
modeling 328-335
partitioning sets 97-98
probe-core plug comparison 255-256
relative 15, 119
truncation 92-94
variability and geology 146-147
permeability relationships
comparisons 203-204
grain size and porosity 14, 120-121
R2 and geology 230-231
p-normal (power-normal) distribution
86, 87
and log-normal distribution 86
and normal distribution 86
population (see sample space)
and exhaustive sequence of events
22, 24
and probability 25, 26
definition 20
porosity
and oil in place 57-58
events 38
histograms and geology 52-54
median 126-128
relationship to density 237,253-254
388
thin section 76
wireline and core plug 186-188, 219222, 223-225
power mean 135-136, 138
probability
addition law of 25
additive and multiplicative rules 95
axioms 25
Bayes Theorem of31-33
conditional26-28, 31-33,39
definition 19, 23
ellipses 196-197
integral81
laws 25·31
multiplication law of 27, 28
of event sequence 25
of events 23-33
plot 81-83
tree 29-31
probability distribution function(s) (PDF)
and geological elements 52
bimodal49
binomial 74-76, 108
chi-squared (x,2) 139, 196
conditionall90-191
continuous 48
definition 43, 47-55
discrete 47
empirical or sample 50
heterogeneous or compound 78
joint 69, 189-191
joint normal (bivariate normal) 195-197
leptokurtic 74
limitations 54
log transformed 50
marginal 69, 190
mesokurtic 74
modeling, estimation and diagnosis of
52
moments of 67, 70
negatively skewed 51, 52
platykurtic 74
positively skewed 53
Index
positively skewed 74
properties of 49
unimodal49, 193
pseudoproperties 344-34 5
random (see stochastic), randomness
apparent 3
definition 2
random variables
continuous 43
definition 37
discrete42
functions of 58
mapping to the real line 37
mixtures 77
two or more 58-60
types 39, 265
reduced major axis (RMA)
and JND regression lines 199-200
and regression 202-203, 237-238
comparing porosity-permeability
relations 202-203
geometric interpretation 201
line of 199-200
variability of intercept for 202
variability of slope for 202
regression
and physically-based models 234
and RMA 199-201, 237-238
ANOV A and 230-232
computer output for 232
elements of 208
geometrical interpretation 201
high leverage points in 226
joint normal PDF lines 199
linear least squares bivariate 209-215
linear model of 198, 209-210
multivariate 257-261
no-intercept model of 232-233
normal equations 212
of a small data set 213-215
parameters of 208
reduction of number of variables
248-251
Statistics for Petroleum Engineers and Geoscientists
sample yon x 235
standard error of216-218
rescaled range 73
residual(s)
analysis 219, 227-228, 242
least-square 211
plots 227, 242
sample space (see also population)
definition 19
sampling
and geology 335-342
biased 110-111
considerations 6, 59, 187-188
effect of correlation 290-291
for semivariogram 281-283
spacing 152, 265
statistical view 57-59, 294
sufficiency 15-154
truncation 89
seismic data 353
semivariogram
definition 270
estimation 276, 285-286
models 288-290
permeability and geology 277-283
sensitivity
analysis 116-119
coefficients 118-122
for Archie's law 116-117
for Carmen-Kozeny equation 119-121
for relative permeabilities 118
shale
detection 27
stochastic 43
simulation
annealing 317-325
averaging methods 316
random residual additions 313-315
sequential gaussian 312-313
sequential indicator 316-317
spectral methods 315
testing 330-335
slope (see also least-squares, regression)
389
confidence interval for 218
estimates of 211-213
independent t-test of 218-221
joint F-test of 222-225
properties of estimates 215-217
standard deviation 74
standard error (see also confidence intervals)
behavior 114
definition 108
of Dykstra-Parsons coefficient 178-180
use in testing 168
standard normal variate 81
and probability plots 82
for logcnormal distribution 84
statistical measures
central tendency 109
correlation 265
dispersion 109
statistics
continuous 41
definition 4
discrete 41
order47
sample or empiricall09
stochastic
definition 2
prediction 2
realization 2
shales 43
variable 37
structure
causality 166
detection and semivariogram 276-283,
13-12
sampling 10
scales 9
variability 146-148
student's t-distribution 132-133
sum of squares 198-199
degrees offreedom for 216
residual216
t-test (student's t-test)
multiple tests 176
Index
390
normality assumption 175-176
of two quantities with equal variances
171-173
of two quantities with unequal variances
174-175
one-sided 175, 185
t-value
critical 173
hypothesis test and computed absolute
171-173
transformations
for joint normality 245-246
from one CDF to another 54
logarithmic 83
power85
regression 241-245
use in tests 176
triangular distribution 88-90
truncated data sets 90-95
bottom 100, 101, 411
correcting 92, 93
top 90, 91, 93, 94, 101
top and bottom 105, 110
type curves for 110
uniform distribution
definition 56, 87
endpoint estimators 112-114
variability
and geology 146-148
and sample size 150-153
measures 144-165
variables (also see random variables)
categorical 39
continuous 38
definition 38-40
dependencies 40
deterministic 39, 40
explanatory 209
fuzzy40
indicator 39
interval38
nominal39, 266
ordinal39
predictor 209
ratiometric 38, 39
response 209
variance
binomial distribution 77
binomial mixture of two random
variables 78
definition 70
discrete 71
exponential distribution 79
Gaussian distribution 80
log-normal distribution 84
minimizing 71
of sample mean 73
of sum of two independent random
variables 72, 73
properties of 71, 72
properties of sample 131-132
sample 71
triangular distribution 91
uniform distribution 78
weighted least squares 247-248
well testing 54, 303-304, 355
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